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Buckling of elastic cylindrical shells considering the effect of localized axisymmetric imperfections
Thin-Walled Structures 42 (2004) 1035–1047 www.elsevier.com/locate/tws
Buckling of elastic cylindrical shells considering the effect of localized axisymmetric imperfections A. Khamlichi a,, M. Bezzazi b, A. Limam c a
Testing Department, Plastex Maroc SA, BP 342 Zone Industrielle, Tangier 90000, Morocco b LGCMS, De´partement de Physique, FSTT, BP 416, Tangier 90000, Morocco c URGC-Structures, INSA de Lyon, 20 Ave Albert Einstein, 69 621 Villeurbanne, France
Received 2 May 2003; received in revised form 17 February 2004; accepted 17 February 2004
Abstract The effect of localized axisymmetric initial imperfections on the critical load of elastic cylindrical shells subjected to axial compression is studied through analytical modeling. Some classical results regarding sensitivity of shell buckling strength with respect to distributed defects having axisymmetric or asymmetric forms are recalled. Special emphasis is placed after that on the more severe case of localized defects satisfying axial symmetry by displaying an analytical solution to the Von Ka´rma´n–Donnell shell equations under specific boundary conditions. The obtained results show that the critical load varies very much with the geometrical parameters of the localized defect. These variations are not monotonic in general. They indicate, however, a clear reduction of the shell critical load for some defects recognized as the most hazardous isolated ones. Reduction of the critical load is found to reach a level which is up to two times lower than that predicted by general distributed defects. # 2004 Elsevier Ltd. All rights reserved. Keywords: Stability; Buckling; Imperfections; Shells; Silos; Localized defects
1. Introduction Thin cylindrical shells, like silos and tanks, continue to be the subject of intensive investigation efforts among researchers. The pursued objectives include the
Corresponding author. Tel.: +212-6779-5068; fax: +212-3935-0702. E-mail address: [email protected] (A. Khamlichi).
0263-8231/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2004.03.008
1036
A. Khamlichi et al. / Thin-Walled Structures 42 (2004) 1035–1047
need to analyse for these special structures, effects of variations affecting shell geometrical parameters on buckling behaviour. These include integrating the effect of initial imperfections in a satisfactory way, especially in the case of welded cylinders. Researchers in this field try to provide rational answers to questions such as: – How does shell strength change with respect to initial geometrical imperfections? – Which imperfection is the most suitable for modeling the knock down factor in a realistic way? – What is the effect of coupling which may arise between various geometrical defects? Geometrical imperfections lead to dispersions, which generally affect the buckling loads obtained from experiments. They are also the main reason for the observed differences between experimental results and theoretical predictions when a perfect shell is assumed. Moreover, if one wants to deal with the problem of material optimisation for shell structures, imperfections must be taken into account in an enhanced quantitative way in order to set up reliable design performances. A complete approach coupling experiments, numerical modeling and imperfection measurements seems to be desirable. Some modern codes of structures, like Eurocode 3, even recommend this process of design optimisation. The buckling behaviour of shell structures depends on interactions which may exist between the different applied loads, because this phenomenon is in essence highly nonlinear. In practice, however, restrictions are made to only reasonable decoupled situations of loading. Complex analyses which take into account the coupling aspect are indeed rare. In the following, attention will be focused on the particular case of a homogenous and isotropic elastic shell subjected to uniform axial compression. Since the pioneer work of Koiter [1], distributed axisymmetric imperfections have been used among other forms of imperfections to study shell sensitivity to initial geometrical defects. Imperfection measurements performed recently by Ding et al. [2] have shown that in particular case of silos, the axisymmetric component of geometrical defects is always preponderant. This result corroborates the well known classical findings of Hutchinson et al. [3], Amazigo and Budiansky [4] and Arbocz [5] regarding the crucial effect of localized axisymmetric imperfections on the buckling load of thin shells. New work presented by Teng and Rotter [6] was dedicated again to shell imperfections of axisymmetric type which may result from welds executed along the circumference of cylinders which are manufactured according to this assembling process. They have undertaken the solution of the resulting equations by means of a special numerical approach. A recent review of the most rigorous developments accomplished in the domain of shell behaviour considering imperfections is presented by the main author Teng [7]. Among the latest contributions regarding this topic, Pircher and Bridge [8] have presented a complete study on the effects induced by circumferential welds on the buckling and post-buckling behaviour of welded cylinders. Interactions between more than one localized defects were considered. Using a commercial finite element code, Kim and
A. Khamlichi et al. / Thin-Walled Structures 42 (2004) 1035–1047
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Kim [9] have once again analyzed the buckling strength of shells having a thickness-to-diameter ratio exceeding 800. Some correlations of the obtained results were performed and design charts were proposed. The authors have outlined the extreme sensitivity to initial imperfections of shells which buckle according to axisymmetric modes. In an attempt to understand the effect of imperfections on the critical load of shells more directly, a general thin cylindrical shell having a circular basis and subjected to quasi-static uniform compressive loads is considered in the following. The assumed shell equations are those of the normal Von Ka´rma´n–Donnell model. Use will also be made of the general analytical equations which were established by Arbocz [5] in order to study the effect of geometrical defects having the usual distributed axisymmetric or asymmetric forms. Considering the particular case of localized axisymmetric defects having the form of tapered belts on the shell circumference, a model for which direct analytical integration can be performed is derived. It is assumed that the defect amplitude is distributed according to a parabolic law over a strip through the shell circumference. This defect constitutes a very particular case among the most general defects which can be modelled by means of numerical integration performed on the ordinary differential system introduced by Arbocz [5]. Even if it is an elementary one, it enables demonstration of the extreme sensitivity of shells to these kinds of localized axisymmetric defects. The results obtained show, in fact, a neat reduction of the critical load. A parametric study has exhibited, for this class of defects, those found to be of particular danger regarding shell buckling strength.
2. The Von Ka´ rma´ n–Donnell equations with geometrical imperfections Earlier studies on buckling of cylindrical elastic shells subjected to axial compressive loads were performed by Timoshenko and Gere [10]. These have been followed by the introduction of more refined and complete shell theories, like those of Von Ka´rma´n–Donnell, Flu¨gge, Sanders and Koiter. Even if these models do not allow one to deal with the problem of shell buckling with the same level of difficulties, they tend altogether to give close results in common situations. In fact, the real problem is not discrepancies between the various theoretical predictions; it is the large gap existing between experimental results and these theoretical predictions. Indeed, it is well known now that this gap is essentially due to geometrical imperfections, which must be taken into account in a satisfactory way. When one seeks for a prompt and efficient solution of shell equations, a pertinent choice is the Von Ka´rma´n–Donnell model, introduced first by Donnell in 1933 to study elastic torsional buckling of circular cylindrical shells. This model has been very attractive and its usage is now almost universal. A version of this model which was introduced in 1950 in order to integrate the effect on buckling due to a small initial perturbation affecting the perfect geometry of the shell structure has recently been used by Arbocz [5]. The Von Ka´rma´n–Donnell shell
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equations have, in this case, the following form: r4 F
Eh 1 w;xx Kðw; w þ 2 wÞ ¼ 0 R 2
Dr4 w þ
h Þ ¼ 0 F;xx hKðF ; w þ w R
ð1Þ ð2Þ
where R is the mean radius of the shell, h is the uniform width of the shell wall, w is the radial shell displacement, which is assumed to be positive if it is directed out is the geometrical defect, F is the Airy function of stresses, ward from the shell, w E is the Young modulus and D is the flexural shell stiffness. x designates the axial coordinate, y is the circumferential coordinate, r4 is the bi-Laplacian operator, and KðX ; Y Þ ¼ X;xx Y;yy 2X;xy Y;xy þ X;yy X;xx . Finally, the comma in subscript position indicates partial differentiation with respect to the quantity which follows it. Fig. 1 shows the shell geometry and summarizes the main notations used. is chosen to have Among the main merits of this model, one may recall that if w the same form of observed post-buckled geometry during experiments, it is possible to theoretically predict the load–displacement curve fitting that obtained from the experiments. Let denote P the axial compressive force acting on the shell, rx the resulting axial stress and L the length of the shell. Arbocz [5] has considered the appropriate
Fig. 1. Geometry of the considered cylindrical shell.
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boundary conditions as given in relations (3) in order to complete Eqs. (1) and (2): wð0; yÞ ¼ wðL; yÞ ¼ w;x ð0; yÞ ¼ w;x ðL; yÞ ¼ 0; P rx ð0; yÞ ¼ rx ðL; yÞ ¼ 2pRh
ð3Þ
and has analyzed a shell defect having the following general form: ðx; yÞ ¼ a0 ðxÞ þ a1 ðxÞcosðny=RÞ w
ð4Þ
where a0 and a1 are arbitrary functions of the axial coordinate x, and n is an integer. Using the Galerkin method, a system of ordinary differential equations is readily obtained. This system can be integrated by the ‘‘shooting method’’ and enables, at least from a theoretical point of view, the study of the effect on shell buckling of any kind of defects having the general form given by Eq. (4). These defects can be either distributed or localized on the shell surface. However, in practice, the operation of integrating a differential system by the ‘‘shooting method’’ suffers the handicap of being very tedious. May be this is why usage of this approach has been so limited in the past and has been only applied to some special cases. For instance, in particular cases such as 1 cos mpx ¼n w ð5Þ L mpx kpx ny ¼ þ n2 sin cos ð6Þ w n1 cos L L R 1 , n 2 are the defect generalized amplitudes, it is possible to simplify the where n original differential system and to search for a solution by simply locating the roots of a polynomial function of degree three. In the following, we proceed by recalling the buckling load for a perfect shell. Then, the particular distributed imperfections having the axisymmetric form (5) or the asymmetric form (6) are introduced, as was first performed by Arbocz [5]. Finally, a localized defect having the form ¼ a0 ðxÞ w
ð7Þ
is considered. In this last case, when the function a0 is parabolic over a narrow strip localized on the shell surface, we demonstrate that an analytical solution exists for the system of Eqs. (1)–(3) and (7). 3. Small deflection theory for a perfect shell If the solution of Eqs. (1) and (2) satisfying boundary conditions (3) is assumed to be of the form mpx ny wðx; yÞ ¼ W sin sin ð8Þ L R where m and n are integers, then one could easily obtain the critical load kc (called
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Fig. 2. Perfect buckling modes corresponding to the classical buckling load.
the classical critical load) as Eh ð9Þ kc ¼ rcx ¼ Rc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where c ¼ 3ð1 t2 Þ. When the shell data are, for example, R ¼ 0:1016 m, L ¼ 0:2032 m, h ¼ 1:179 10-4 m, E ¼ 1:048 1011 Pa, and t ¼ 0:3, one finds that kc ¼ 7:36 107 Pa. The previous data ensure that buckling occurs in the elastic domain. The obtained critical load is associated to a set of buckling modes, as shown in Fig. 2. Particular buckling modes are given, for example, by: (m ¼ 2, n ¼ 26), (m ¼ 3, n ¼ 30), (m ¼ 5, n ¼ 40). This is a feature of the Von Ka´rma´n–Donnell model, which does not allow selection of buckling modes in case of a perfect shell. In constant use of refined shell theories like those of Flu¨gge or Koiter results in a unique buckling mode, which is, in the present case, the mode (m ¼ 2, n ¼ 26). This is not, however, an authentic drawback of the Von Ka´rma´n–Donnell model, since the real problem in using Eq. (9) is that the results obtained are always in complete disagreement with the buckling loads determined from experiments such as those due, for example, to Lundquist [11] and Weingarten et al. [12].
4. Effect of distributed axisymmetric imperfections These kinds of defects are essentially due to the manufacturing processes of the cylindrical shells. Their simplest representation is a sinusoidal axial wave having the form given by relation (5) with m an even integer. The solution is assumed to admit the form mpx ny w ¼ n1 sin cos ð10Þ 2L R
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Fig. 3. Reduced buckling load (axisymmetric distributed defects).
and the critical load k as a function of the defect magnitude n1 , and m is calculated by means of a numerical program. For this purpose, a minimisation with respect to the number of circumferential half waves n is performed for any given shell parameters. When the shell data are the same as those previously used for a perfect shell, one finds the buckling loads and modes shown in Figs. 3 and 4.
5. Effect of distributed asymmetric imperfections An asymmetric defect having the form given by Eq. (6), with k ¼ m=2 and m an even integer, has the same axisymmetric component as the defect defined by relation (5). This is an interesting defect since it describes some of the post-buckled
Fig. 4. Buckling modes (axisymmetric distributed defects).
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shapes obtained during experimentation. If the pre-buckling solution is assumed to be of the form mpx kpx ny þ n2 sin cos ; w ¼ n0 þ n1 cos L L R F ¼
Eh2 k 2 y þ f0 ðxÞ 2cR
ð11Þ
Arbocz [5] had shown that the solution must satisfy a system of two polynomial equations of unknowns n1 and n2. By fixing the value of n2, the solution can be considered in the plane (k, n1). This simplifies the previous system to a single polynomial equation of degree three. The critical load k is the positive minimum obtained from the three roots corresponding to the previous equation. Now representing k versus n2 enables location of the critical load, which is given by the first reached maximum kd (limit point) of this last curve when it exists. A numerical program was written in order to automatically compute the critical load by performing (i) a minimisation with respect to n in order to evaluate the number of circumferential half waves fixing the buckling mode and (ii) a second minimisation with respect to the number of axial half waves m. Table 1 gives the buckling mode m and the critical load corresponding to four different cases of asymmetric defects. The data used in this application are the same as those used for the perfect shell.
6. Effect of localized axisymmetric imperfections When a localized defect is assumed to have the form given by Eq. (7), the differential system obtained by Arbocz [5] reduces to the following form: ðw0 Þ;zzzz þ 2aðw0 Þ;zz þ bw0 ¼ 2aða0 Þ;zz
ð12Þ
where z ¼ x=R, a ¼ 2Rkc=h and b ¼ 4c2 R2 =h2 . It can be shown, on the other hand, that the boundary conditions as given by Eq. (3) are easily transformed to the form ^¼ w0 ð0Þ ¼ w0 ðL=RÞ ¼ w
tk ; and ðw0 Þ;z ð0Þ ¼ ðw0 Þ;z ðL=RÞ ¼ 0 c
Table 1 Buckling load versus the amplitude of the distributed asymmetric imperfections
Case 1 Case 2 Case 3 Case 4
n1 (106 m)
n2 (106 m)
m
kd =kc
11.79 47.16 47.16 11.79
11.79 11.79 47.16 47.16
16 14 8 10
0.9749 0.8360 0.2876 0.3329
ð13Þ
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Given an axisymmetric defect having the form of a parabola of magnitude e which is distributed on the strip ½ðx0 gÞ=2; ðx0 þ gÞ=2 , performing analytic integration of Eqs. (12) and (13) yields the following expression of shell radial displacement: ^ ðkÞ þ w ~ ðk; e; g; x0 ÞÞ þ ht Sðk; z; e; g; x0 ÞGðzÞ wðxÞ ¼ hðw
ð14Þ
where S ¼ T 1 W , t GðzÞ ¼ ½ euz cosðwzÞ euz sinðwzÞ euz cosðwzÞ euz sinðwzÞ , ~ ðk; e; g; x0 Þ ¼ 8h2 Rek=cgðx0 þ gÞ, and t in superscript position indicates the matrix w transpose. In the previous relations, the other notations used are 3 2 1 0 1 0 7 6 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 u w u w 7 6 cR 7 6 ð1 kÞ; T ¼6 7; u ¼ h 7 6 s1 s2 s3 s4 5 4 us1 ws2 us2 þ ws1 us3 ws4 us4 þ ws3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cR w¼ ð1 þ kÞ; s1 ¼ euL=R cosðwL=RÞ; euL=R sinðwL=RÞ; h ~ Þ 0 ð^ ~Þ 0
wþw wþw euL=R cosðwL=RÞ; euL=R sinðwL=RÞ and t W ¼ ½ ð^ Parametric studies regarding shell buckling can now be performed. Using the same shell data as previously considered for the perfect shell, Figs. 5 and 6 give predictions of the buckling load for the centred localized defects as a function of defect characteristics amplitude e and strip width g. To obtain these figures, the load– deflection curve as given by relation (14) is plotted. Then the first bifurcation or limit point when it occurs is located. Fig. 7 shows typical bifurcation points in the load–deflection curves. Fig. 8 shows the buckling modes associated with the previous buckling loads.
Fig. 5. Buckling load versus the defect amplitude |e| for g ¼ L=20.
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Fig. 6. Buckling load versus the defect strip width g for jej ¼ h=10.
7. Comparison with other known results In order to evaluate the range of validity of the actual buckling strength predictions, let us first consider the governing sensitivity expression for the relative buckling load versus the amplitude of a distributed axisymmetric imperfection as established for long cylindrical shells by Hutchinson and Koiter [13]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 3c 3c 9c2 2 eþ ð15Þ ¼1þ e e kc 4h 2h 16h2 where k is the actual buckling load when a geometrical imperfection in the shape of the axisymmetric buckling mode of amplitude e is assumed.
Fig. 7. Load–deflection curves as a function of |e| for g ¼ L=20.
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Fig. 8. Shell buckling modes for two defect amplitudes and g ¼ L=20.
To also assess comparisons with the case of distributed defects obtained by linear combination of the first buckling modes, the numerical results obtained through use of a commercial finite element code by Kim and Kim [9] are also considered. Table 2 summarizes the obtained buckling load as a function of the defect amplitude. The boundary conditions used in obtaining the numerical results are those corresponding to a cylinder on a stiff foundation, which is close to the shell considered in the present work. The authors have outlined that the results are scarcely affected by the boundary conditions, since when considering the same cylinder on a soft foundation, variations of the results were limited to within a 6% interval. In fact, the results associated to shell parameters bounding the case L ¼ 2R et R ¼ 866 h considered in the present work are recalled in Table 2. As shown in Table 2, our model underestimates the buckling load when the localized defect amplitude is very small: jej ¼ 0:1 h. Our model nevertheless predicts nearly the same buckling load for moderate defect amplitudes, jej 0:3 h, as that obtained from Kim and Kim [9], while it overestimates the buckling load in
Table 2 Comparison of reduced buckling loads jej=h
L ¼ 2R and R ¼ 800h [9]
L ¼ 2R and R ¼ 1000h [9]
L ¼ 2R and R ¼ 866h (Hutchinson et al.)
L ¼ 2R and R ¼ 866h (this study)
0.1 0.2 0.3 1 2 3
0.608 n.c. 0.420 0.310 0.257 0.231
0.695 n.c. 0.532 0.328 0.301 0.262
0.611 0.501 0.433 0.236 0.147 0.107
0.978 0.110 0.224 0.271 0.280 0.282
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comparison with Hutchinson and Koiter [13]. Note, however, that our model predicts, in the most severe case, a critical load which is only 0.110 when jej ¼ 0:2 h. This result is only one half the critical load predicted by Kim and Kim [9] and agrees with the minimum buckling load obtained by Hutchinson and Koiter [13] when jej ¼ 3 h. Note also that the governing sensitivity expression given in Eq. (15) predicts a buckling load which is always decreasing while the localized imperfection considered here predicts a sensitivity behaviour to defect amplitude which is not monotonic.
8. Discussions and conclusions When the shell surface suffers from the presence of a localized axisymmetric imperfection, the fundamental equilibrium trajectory bifurcates or admits a limit point. The actual critical load is less than the classical critical load. Figs. 5 and 6 and Table 2 show that the obtained critical loads can be much less in comparison with those obtained by distributed defects. It is possible to perform, by means of the analytic model presented in this work, parametric studies with the view of locating defects which are potentially the most dangerous. In all cases, calculations are achieved with high accuracy and speed. It has been shown that the effect of a localized axisymmetric defect is not proportional to its span. Moreover, it is not possible to foresee which one would have the most important effect on the buckling load. Considering each possible case separately is hence necessary. In practice, one may get the actual imperfections by taking measurements and introduce them in the analytic model after performing a correlation with the localized defect having the form considered in this work. In the absence of any imperfection statements, it is advisable to consider the most adverse critical load obtained from various simulations. This last could be obtained, in fact, through a minimisation process conducted over the parameters of the localized axisymmetric defect: amplitude and strip width.
References [1] Koiter WT. On the stability of elastic equilibrium. Ph.D. Thesis. Technological University of Delft, Holland, 1945 [English translation issued in 1967 as NASA, Tech Trans F10, 833]. [2] Ding X, Coleman R, Rotter JM. Technique for precise measurement of large-scale silos and tanks. Journal of Surveying Engineering 1996;122(1):15–25. [3] Hutchinson JW, Tennyson RC, Muggeridge DB. Effect of local axisymmetric imperfection on the buckling of a cylindrical shell under axial compression. AIAA Journal 1972;9(1):48–52. [4] Amazigo JC, Budiansky B. Asymptotic formulas for the buckling stresses of axially compressed cylinders with localized or random axisymmetric imperfections. Journal of Applied Mechanics 1972;93:179–84. [5] Arbocz J. Post-buckling behaviour of structures, numerical techniques for more complicated structures. Buckling and post-buckling of structures. Berlin, Heidelberg: Springer-Verlag; 1987, p. 83–142. [6] Teng JC, Rotter JM. Buckling of pressurized axisymmetrically imperfect cylinders under axial loads. Journal of Engineering Mechanics 1992;118(2):229–47.
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[7] Teng JC. Buckling of thins shells: recent advances and trends. Journal Mechanics Reviews ASME 1996;49(4):263–74. [8] Pircher M, Bridge RQ. The influence of circumferential weld-induced imperfections on the buckling of silos and tanks. Journal of Constructional Steel Research 2001;57:569–80. [9] Kim SE, Kim CS. Buckling strength of the cylindrical shell and tank subjected to axially compressive loads. Thin-Walled Structures 2002;40:329–53. [10] Timoshenko SP, Gere JM. Theory of elastic stability. New York: Mc Graw-Hill; 1961. [11] Lundquist EE. Strength tests of thin-walled duralumin cylinders in compression. NACA Report No. 473, 1933. [12] Weingarten VI, Morgan EJ, Seide P. Elastic stability of thin-walled cylindrical and conical shells under axial compression. AIAA Journal 1965;3:500–55. [13] Hutchinson JW, Koiter WT. Post-buckling theory. Applied Mechanical Review 1970;23:1353–66.
Buckling of elastic cylindrical shells considering the effect of localized axisymmetric imperfections A. Khamlichi a,, M. Bezzazi b, A. Limam c a
Testing Department, Plastex Maroc SA, BP 342 Zone Industrielle, Tangier 90000, Morocco b LGCMS, De´partement de Physique, FSTT, BP 416, Tangier 90000, Morocco c URGC-Structures, INSA de Lyon, 20 Ave Albert Einstein, 69 621 Villeurbanne, France
Received 2 May 2003; received in revised form 17 February 2004; accepted 17 February 2004
Abstract The effect of localized axisymmetric initial imperfections on the critical load of elastic cylindrical shells subjected to axial compression is studied through analytical modeling. Some classical results regarding sensitivity of shell buckling strength with respect to distributed defects having axisymmetric or asymmetric forms are recalled. Special emphasis is placed after that on the more severe case of localized defects satisfying axial symmetry by displaying an analytical solution to the Von Ka´rma´n–Donnell shell equations under specific boundary conditions. The obtained results show that the critical load varies very much with the geometrical parameters of the localized defect. These variations are not monotonic in general. They indicate, however, a clear reduction of the shell critical load for some defects recognized as the most hazardous isolated ones. Reduction of the critical load is found to reach a level which is up to two times lower than that predicted by general distributed defects. # 2004 Elsevier Ltd. All rights reserved. Keywords: Stability; Buckling; Imperfections; Shells; Silos; Localized defects
1. Introduction Thin cylindrical shells, like silos and tanks, continue to be the subject of intensive investigation efforts among researchers. The pursued objectives include the
Corresponding author. Tel.: +212-6779-5068; fax: +212-3935-0702. E-mail address: [email protected] (A. Khamlichi).
0263-8231/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2004.03.008
1036
A. Khamlichi et al. / Thin-Walled Structures 42 (2004) 1035–1047
need to analyse for these special structures, effects of variations affecting shell geometrical parameters on buckling behaviour. These include integrating the effect of initial imperfections in a satisfactory way, especially in the case of welded cylinders. Researchers in this field try to provide rational answers to questions such as: – How does shell strength change with respect to initial geometrical imperfections? – Which imperfection is the most suitable for modeling the knock down factor in a realistic way? – What is the effect of coupling which may arise between various geometrical defects? Geometrical imperfections lead to dispersions, which generally affect the buckling loads obtained from experiments. They are also the main reason for the observed differences between experimental results and theoretical predictions when a perfect shell is assumed. Moreover, if one wants to deal with the problem of material optimisation for shell structures, imperfections must be taken into account in an enhanced quantitative way in order to set up reliable design performances. A complete approach coupling experiments, numerical modeling and imperfection measurements seems to be desirable. Some modern codes of structures, like Eurocode 3, even recommend this process of design optimisation. The buckling behaviour of shell structures depends on interactions which may exist between the different applied loads, because this phenomenon is in essence highly nonlinear. In practice, however, restrictions are made to only reasonable decoupled situations of loading. Complex analyses which take into account the coupling aspect are indeed rare. In the following, attention will be focused on the particular case of a homogenous and isotropic elastic shell subjected to uniform axial compression. Since the pioneer work of Koiter [1], distributed axisymmetric imperfections have been used among other forms of imperfections to study shell sensitivity to initial geometrical defects. Imperfection measurements performed recently by Ding et al. [2] have shown that in particular case of silos, the axisymmetric component of geometrical defects is always preponderant. This result corroborates the well known classical findings of Hutchinson et al. [3], Amazigo and Budiansky [4] and Arbocz [5] regarding the crucial effect of localized axisymmetric imperfections on the buckling load of thin shells. New work presented by Teng and Rotter [6] was dedicated again to shell imperfections of axisymmetric type which may result from welds executed along the circumference of cylinders which are manufactured according to this assembling process. They have undertaken the solution of the resulting equations by means of a special numerical approach. A recent review of the most rigorous developments accomplished in the domain of shell behaviour considering imperfections is presented by the main author Teng [7]. Among the latest contributions regarding this topic, Pircher and Bridge [8] have presented a complete study on the effects induced by circumferential welds on the buckling and post-buckling behaviour of welded cylinders. Interactions between more than one localized defects were considered. Using a commercial finite element code, Kim and
A. Khamlichi et al. / Thin-Walled Structures 42 (2004) 1035–1047
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Kim [9] have once again analyzed the buckling strength of shells having a thickness-to-diameter ratio exceeding 800. Some correlations of the obtained results were performed and design charts were proposed. The authors have outlined the extreme sensitivity to initial imperfections of shells which buckle according to axisymmetric modes. In an attempt to understand the effect of imperfections on the critical load of shells more directly, a general thin cylindrical shell having a circular basis and subjected to quasi-static uniform compressive loads is considered in the following. The assumed shell equations are those of the normal Von Ka´rma´n–Donnell model. Use will also be made of the general analytical equations which were established by Arbocz [5] in order to study the effect of geometrical defects having the usual distributed axisymmetric or asymmetric forms. Considering the particular case of localized axisymmetric defects having the form of tapered belts on the shell circumference, a model for which direct analytical integration can be performed is derived. It is assumed that the defect amplitude is distributed according to a parabolic law over a strip through the shell circumference. This defect constitutes a very particular case among the most general defects which can be modelled by means of numerical integration performed on the ordinary differential system introduced by Arbocz [5]. Even if it is an elementary one, it enables demonstration of the extreme sensitivity of shells to these kinds of localized axisymmetric defects. The results obtained show, in fact, a neat reduction of the critical load. A parametric study has exhibited, for this class of defects, those found to be of particular danger regarding shell buckling strength.
2. The Von Ka´ rma´ n–Donnell equations with geometrical imperfections Earlier studies on buckling of cylindrical elastic shells subjected to axial compressive loads were performed by Timoshenko and Gere [10]. These have been followed by the introduction of more refined and complete shell theories, like those of Von Ka´rma´n–Donnell, Flu¨gge, Sanders and Koiter. Even if these models do not allow one to deal with the problem of shell buckling with the same level of difficulties, they tend altogether to give close results in common situations. In fact, the real problem is not discrepancies between the various theoretical predictions; it is the large gap existing between experimental results and these theoretical predictions. Indeed, it is well known now that this gap is essentially due to geometrical imperfections, which must be taken into account in a satisfactory way. When one seeks for a prompt and efficient solution of shell equations, a pertinent choice is the Von Ka´rma´n–Donnell model, introduced first by Donnell in 1933 to study elastic torsional buckling of circular cylindrical shells. This model has been very attractive and its usage is now almost universal. A version of this model which was introduced in 1950 in order to integrate the effect on buckling due to a small initial perturbation affecting the perfect geometry of the shell structure has recently been used by Arbocz [5]. The Von Ka´rma´n–Donnell shell
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equations have, in this case, the following form: r4 F
Eh 1 w;xx Kðw; w þ 2 wÞ ¼ 0 R 2
Dr4 w þ
h Þ ¼ 0 F;xx hKðF ; w þ w R
ð1Þ ð2Þ
where R is the mean radius of the shell, h is the uniform width of the shell wall, w is the radial shell displacement, which is assumed to be positive if it is directed out is the geometrical defect, F is the Airy function of stresses, ward from the shell, w E is the Young modulus and D is the flexural shell stiffness. x designates the axial coordinate, y is the circumferential coordinate, r4 is the bi-Laplacian operator, and KðX ; Y Þ ¼ X;xx Y;yy 2X;xy Y;xy þ X;yy X;xx . Finally, the comma in subscript position indicates partial differentiation with respect to the quantity which follows it. Fig. 1 shows the shell geometry and summarizes the main notations used. is chosen to have Among the main merits of this model, one may recall that if w the same form of observed post-buckled geometry during experiments, it is possible to theoretically predict the load–displacement curve fitting that obtained from the experiments. Let denote P the axial compressive force acting on the shell, rx the resulting axial stress and L the length of the shell. Arbocz [5] has considered the appropriate
Fig. 1. Geometry of the considered cylindrical shell.
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boundary conditions as given in relations (3) in order to complete Eqs. (1) and (2): wð0; yÞ ¼ wðL; yÞ ¼ w;x ð0; yÞ ¼ w;x ðL; yÞ ¼ 0; P rx ð0; yÞ ¼ rx ðL; yÞ ¼ 2pRh
ð3Þ
and has analyzed a shell defect having the following general form: ðx; yÞ ¼ a0 ðxÞ þ a1 ðxÞcosðny=RÞ w
ð4Þ
where a0 and a1 are arbitrary functions of the axial coordinate x, and n is an integer. Using the Galerkin method, a system of ordinary differential equations is readily obtained. This system can be integrated by the ‘‘shooting method’’ and enables, at least from a theoretical point of view, the study of the effect on shell buckling of any kind of defects having the general form given by Eq. (4). These defects can be either distributed or localized on the shell surface. However, in practice, the operation of integrating a differential system by the ‘‘shooting method’’ suffers the handicap of being very tedious. May be this is why usage of this approach has been so limited in the past and has been only applied to some special cases. For instance, in particular cases such as 1 cos mpx ¼n w ð5Þ L mpx kpx ny ¼ þ n2 sin cos ð6Þ w n1 cos L L R 1 , n 2 are the defect generalized amplitudes, it is possible to simplify the where n original differential system and to search for a solution by simply locating the roots of a polynomial function of degree three. In the following, we proceed by recalling the buckling load for a perfect shell. Then, the particular distributed imperfections having the axisymmetric form (5) or the asymmetric form (6) are introduced, as was first performed by Arbocz [5]. Finally, a localized defect having the form ¼ a0 ðxÞ w
ð7Þ
is considered. In this last case, when the function a0 is parabolic over a narrow strip localized on the shell surface, we demonstrate that an analytical solution exists for the system of Eqs. (1)–(3) and (7). 3. Small deflection theory for a perfect shell If the solution of Eqs. (1) and (2) satisfying boundary conditions (3) is assumed to be of the form mpx ny wðx; yÞ ¼ W sin sin ð8Þ L R where m and n are integers, then one could easily obtain the critical load kc (called
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Fig. 2. Perfect buckling modes corresponding to the classical buckling load.
the classical critical load) as Eh ð9Þ kc ¼ rcx ¼ Rc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where c ¼ 3ð1 t2 Þ. When the shell data are, for example, R ¼ 0:1016 m, L ¼ 0:2032 m, h ¼ 1:179 10-4 m, E ¼ 1:048 1011 Pa, and t ¼ 0:3, one finds that kc ¼ 7:36 107 Pa. The previous data ensure that buckling occurs in the elastic domain. The obtained critical load is associated to a set of buckling modes, as shown in Fig. 2. Particular buckling modes are given, for example, by: (m ¼ 2, n ¼ 26), (m ¼ 3, n ¼ 30), (m ¼ 5, n ¼ 40). This is a feature of the Von Ka´rma´n–Donnell model, which does not allow selection of buckling modes in case of a perfect shell. In constant use of refined shell theories like those of Flu¨gge or Koiter results in a unique buckling mode, which is, in the present case, the mode (m ¼ 2, n ¼ 26). This is not, however, an authentic drawback of the Von Ka´rma´n–Donnell model, since the real problem in using Eq. (9) is that the results obtained are always in complete disagreement with the buckling loads determined from experiments such as those due, for example, to Lundquist [11] and Weingarten et al. [12].
4. Effect of distributed axisymmetric imperfections These kinds of defects are essentially due to the manufacturing processes of the cylindrical shells. Their simplest representation is a sinusoidal axial wave having the form given by relation (5) with m an even integer. The solution is assumed to admit the form mpx ny w ¼ n1 sin cos ð10Þ 2L R
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Fig. 3. Reduced buckling load (axisymmetric distributed defects).
and the critical load k as a function of the defect magnitude n1 , and m is calculated by means of a numerical program. For this purpose, a minimisation with respect to the number of circumferential half waves n is performed for any given shell parameters. When the shell data are the same as those previously used for a perfect shell, one finds the buckling loads and modes shown in Figs. 3 and 4.
5. Effect of distributed asymmetric imperfections An asymmetric defect having the form given by Eq. (6), with k ¼ m=2 and m an even integer, has the same axisymmetric component as the defect defined by relation (5). This is an interesting defect since it describes some of the post-buckled
Fig. 4. Buckling modes (axisymmetric distributed defects).
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shapes obtained during experimentation. If the pre-buckling solution is assumed to be of the form mpx kpx ny þ n2 sin cos ; w ¼ n0 þ n1 cos L L R F ¼
Eh2 k 2 y þ f0 ðxÞ 2cR
ð11Þ
Arbocz [5] had shown that the solution must satisfy a system of two polynomial equations of unknowns n1 and n2. By fixing the value of n2, the solution can be considered in the plane (k, n1). This simplifies the previous system to a single polynomial equation of degree three. The critical load k is the positive minimum obtained from the three roots corresponding to the previous equation. Now representing k versus n2 enables location of the critical load, which is given by the first reached maximum kd (limit point) of this last curve when it exists. A numerical program was written in order to automatically compute the critical load by performing (i) a minimisation with respect to n in order to evaluate the number of circumferential half waves fixing the buckling mode and (ii) a second minimisation with respect to the number of axial half waves m. Table 1 gives the buckling mode m and the critical load corresponding to four different cases of asymmetric defects. The data used in this application are the same as those used for the perfect shell.
6. Effect of localized axisymmetric imperfections When a localized defect is assumed to have the form given by Eq. (7), the differential system obtained by Arbocz [5] reduces to the following form: ðw0 Þ;zzzz þ 2aðw0 Þ;zz þ bw0 ¼ 2aða0 Þ;zz
ð12Þ
where z ¼ x=R, a ¼ 2Rkc=h and b ¼ 4c2 R2 =h2 . It can be shown, on the other hand, that the boundary conditions as given by Eq. (3) are easily transformed to the form ^¼ w0 ð0Þ ¼ w0 ðL=RÞ ¼ w
tk ; and ðw0 Þ;z ð0Þ ¼ ðw0 Þ;z ðL=RÞ ¼ 0 c
Table 1 Buckling load versus the amplitude of the distributed asymmetric imperfections
Case 1 Case 2 Case 3 Case 4
n1 (106 m)
n2 (106 m)
m
kd =kc
11.79 47.16 47.16 11.79
11.79 11.79 47.16 47.16
16 14 8 10
0.9749 0.8360 0.2876 0.3329
ð13Þ
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Given an axisymmetric defect having the form of a parabola of magnitude e which is distributed on the strip ½ðx0 gÞ=2; ðx0 þ gÞ=2 , performing analytic integration of Eqs. (12) and (13) yields the following expression of shell radial displacement: ^ ðkÞ þ w ~ ðk; e; g; x0 ÞÞ þ ht Sðk; z; e; g; x0 ÞGðzÞ wðxÞ ¼ hðw
ð14Þ
where S ¼ T 1 W , t GðzÞ ¼ ½ euz cosðwzÞ euz sinðwzÞ euz cosðwzÞ euz sinðwzÞ , ~ ðk; e; g; x0 Þ ¼ 8h2 Rek=cgðx0 þ gÞ, and t in superscript position indicates the matrix w transpose. In the previous relations, the other notations used are 3 2 1 0 1 0 7 6 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 u w u w 7 6 cR 7 6 ð1 kÞ; T ¼6 7; u ¼ h 7 6 s1 s2 s3 s4 5 4 us1 ws2 us2 þ ws1 us3 ws4 us4 þ ws3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cR w¼ ð1 þ kÞ; s1 ¼ euL=R cosðwL=RÞ; euL=R sinðwL=RÞ; h ~ Þ 0 ð^ ~Þ 0
wþw wþw euL=R cosðwL=RÞ; euL=R sinðwL=RÞ and t W ¼ ½ ð^ Parametric studies regarding shell buckling can now be performed. Using the same shell data as previously considered for the perfect shell, Figs. 5 and 6 give predictions of the buckling load for the centred localized defects as a function of defect characteristics amplitude e and strip width g. To obtain these figures, the load– deflection curve as given by relation (14) is plotted. Then the first bifurcation or limit point when it occurs is located. Fig. 7 shows typical bifurcation points in the load–deflection curves. Fig. 8 shows the buckling modes associated with the previous buckling loads.
Fig. 5. Buckling load versus the defect amplitude |e| for g ¼ L=20.
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Fig. 6. Buckling load versus the defect strip width g for jej ¼ h=10.
7. Comparison with other known results In order to evaluate the range of validity of the actual buckling strength predictions, let us first consider the governing sensitivity expression for the relative buckling load versus the amplitude of a distributed axisymmetric imperfection as established for long cylindrical shells by Hutchinson and Koiter [13]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 3c 3c 9c2 2 eþ ð15Þ ¼1þ e e kc 4h 2h 16h2 where k is the actual buckling load when a geometrical imperfection in the shape of the axisymmetric buckling mode of amplitude e is assumed.
Fig. 7. Load–deflection curves as a function of |e| for g ¼ L=20.
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Fig. 8. Shell buckling modes for two defect amplitudes and g ¼ L=20.
To also assess comparisons with the case of distributed defects obtained by linear combination of the first buckling modes, the numerical results obtained through use of a commercial finite element code by Kim and Kim [9] are also considered. Table 2 summarizes the obtained buckling load as a function of the defect amplitude. The boundary conditions used in obtaining the numerical results are those corresponding to a cylinder on a stiff foundation, which is close to the shell considered in the present work. The authors have outlined that the results are scarcely affected by the boundary conditions, since when considering the same cylinder on a soft foundation, variations of the results were limited to within a 6% interval. In fact, the results associated to shell parameters bounding the case L ¼ 2R et R ¼ 866 h considered in the present work are recalled in Table 2. As shown in Table 2, our model underestimates the buckling load when the localized defect amplitude is very small: jej ¼ 0:1 h. Our model nevertheless predicts nearly the same buckling load for moderate defect amplitudes, jej 0:3 h, as that obtained from Kim and Kim [9], while it overestimates the buckling load in
Table 2 Comparison of reduced buckling loads jej=h
L ¼ 2R and R ¼ 800h [9]
L ¼ 2R and R ¼ 1000h [9]
L ¼ 2R and R ¼ 866h (Hutchinson et al.)
L ¼ 2R and R ¼ 866h (this study)
0.1 0.2 0.3 1 2 3
0.608 n.c. 0.420 0.310 0.257 0.231
0.695 n.c. 0.532 0.328 0.301 0.262
0.611 0.501 0.433 0.236 0.147 0.107
0.978 0.110 0.224 0.271 0.280 0.282
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comparison with Hutchinson and Koiter [13]. Note, however, that our model predicts, in the most severe case, a critical load which is only 0.110 when jej ¼ 0:2 h. This result is only one half the critical load predicted by Kim and Kim [9] and agrees with the minimum buckling load obtained by Hutchinson and Koiter [13] when jej ¼ 3 h. Note also that the governing sensitivity expression given in Eq. (15) predicts a buckling load which is always decreasing while the localized imperfection considered here predicts a sensitivity behaviour to defect amplitude which is not monotonic.
8. Discussions and conclusions When the shell surface suffers from the presence of a localized axisymmetric imperfection, the fundamental equilibrium trajectory bifurcates or admits a limit point. The actual critical load is less than the classical critical load. Figs. 5 and 6 and Table 2 show that the obtained critical loads can be much less in comparison with those obtained by distributed defects. It is possible to perform, by means of the analytic model presented in this work, parametric studies with the view of locating defects which are potentially the most dangerous. In all cases, calculations are achieved with high accuracy and speed. It has been shown that the effect of a localized axisymmetric defect is not proportional to its span. Moreover, it is not possible to foresee which one would have the most important effect on the buckling load. Considering each possible case separately is hence necessary. In practice, one may get the actual imperfections by taking measurements and introduce them in the analytic model after performing a correlation with the localized defect having the form considered in this work. In the absence of any imperfection statements, it is advisable to consider the most adverse critical load obtained from various simulations. This last could be obtained, in fact, through a minimisation process conducted over the parameters of the localized axisymmetric defect: amplitude and strip width.
References [1] Koiter WT. On the stability of elastic equilibrium. Ph.D. Thesis. Technological University of Delft, Holland, 1945 [English translation issued in 1967 as NASA, Tech Trans F10, 833]. [2] Ding X, Coleman R, Rotter JM. Technique for precise measurement of large-scale silos and tanks. Journal of Surveying Engineering 1996;122(1):15–25. [3] Hutchinson JW, Tennyson RC, Muggeridge DB. Effect of local axisymmetric imperfection on the buckling of a cylindrical shell under axial compression. AIAA Journal 1972;9(1):48–52. [4] Amazigo JC, Budiansky B. Asymptotic formulas for the buckling stresses of axially compressed cylinders with localized or random axisymmetric imperfections. Journal of Applied Mechanics 1972;93:179–84. [5] Arbocz J. Post-buckling behaviour of structures, numerical techniques for more complicated structures. Buckling and post-buckling of structures. Berlin, Heidelberg: Springer-Verlag; 1987, p. 83–142. [6] Teng JC, Rotter JM. Buckling of pressurized axisymmetrically imperfect cylinders under axial loads. Journal of Engineering Mechanics 1992;118(2):229–47.
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[7] Teng JC. Buckling of thins shells: recent advances and trends. Journal Mechanics Reviews ASME 1996;49(4):263–74. [8] Pircher M, Bridge RQ. The influence of circumferential weld-induced imperfections on the buckling of silos and tanks. Journal of Constructional Steel Research 2001;57:569–80. [9] Kim SE, Kim CS. Buckling strength of the cylindrical shell and tank subjected to axially compressive loads. Thin-Walled Structures 2002;40:329–53. [10] Timoshenko SP, Gere JM. Theory of elastic stability. New York: Mc Graw-Hill; 1961. [11] Lundquist EE. Strength tests of thin-walled duralumin cylinders in compression. NACA Report No. 473, 1933. [12] Weingarten VI, Morgan EJ, Seide P. Elastic stability of thin-walled cylindrical and conical shells under axial compression. AIAA Journal 1965;3:500–55. [13] Hutchinson JW, Koiter WT. Post-buckling theory. Applied Mechanical Review 1970;23:1353–66.