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An analytical study of buckling of composite tubes with various boundary conditions
Composite StructuresVol. 39, No. l-2, pp. 157-164, 1997 0 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0263~8223/97/$17.00 + 0.00 ELSEVIER
PII:SO263-8223(97)00135-9
An analytical study of buckling of composite tubes with various boundary conditions S. T. S. Al-Hassani, M. Darvizeh* & H. Haftchenari Applied Mechanics Division, Mechanical Engineering Dept., UMIST PO Box 88, Sackville Street, Manchester M60 lQD, UK
A theoretical investigation of the buckling problem of composite tubes with different boundary conditions is presented. Different types of loading such as compressive, torsional, internal, external or any combination of them can be applied. To increase the accuracy of results, the modal forms are assumed to have axial dependency in the form of simple Fourier series. The derivatives of the Fourier series are legitimised using Stoke’s transformation. Fhigge’s shell equations, modified for anisotropic laminated materials, are used. The method developed in this work for finding the buckling loads yields an exact equation which is simpler than the exact method adopted by other research workers. To check the validity of the present analysis, the results are compared with some available theoretical and experimental results. 0 1997 Elsevier Science Ltd.
NOMENCLATURE
INTRODUCTION
The buckling of fibrous composite structures with different boundary conditions and various types of loading is of particular interest to engineers in modern technology. This is due to the importance of basic load carrying members of advanced structures such as aircraft fuselages, submarine hulls, booster rockets, pipelines and pressure vessels. The introduction of high strength to weight ratio of fibre-reinforced composite material has led to greater flexibility in the design of advanced structures. As a result there has been a considerable interest in developing a viable testing technique to characterise the mechanical properties of composites. It has been realised that testing of tubular specimens may lead to more reliable results and some problems such as end conditions and other implications in testing procedures are less significant compared with tests on coupon type specimens where end gripping exists. A typical problem with testing tubular specimens is the presence of buckling. Designers and experimentalists are often interested in a quick and accurate estimate of the buckling load which normally requires a comprehensive development of an exact mathematical model.
elastic moduli of the kth laminate shell stiffnesses Fourier coefficients shell diameter transverse modulus matrix elements thickness of the kth shell layer total shell thickness shell length axial mode number circumferential mode number shell radius N,(O, B)/cos ntl -N,(L, B)/cos ne - 7cbiyo,ql(L cos pq
nM,(L,ey(~ cos ne) in-plane and radial displacement components axial and circumferential coordinates
*On leave from the Department of Mechanical Engineering, University of Guilan, PO Box 401, Rasht, Iran.
157
158
S. 7: S. Al-Hassani, M. Darvizeh, H. Haftchenati
In Ref. 1 the plastic buckling of thin-walled aluminium tubes and rings of different sizes, under intense external magnetic pressure has been investigated. The equations related to the radial buckling analysis given by Ref. 2 have also been developed. In Ref. 3, theoretical results using exact and approximate methods are presented for shells under axial compression and torsion with clamped-clamped boundary conditions. Experimental results concerning the failure prediction of filament wound glass reinforced plastics are presented in Ref. 4. In Ref. 5, the finite element method is used to evaluate the effects of in-plane shear non-linearity on buckling response of clamped-clamped shells. A theoretical study of the response of filament wound composite shells with clamped ends under internal pressure is presented in Ref. 6. In Ref. 7, the buckling of elastoplastic circular cylindrical shells with clamped ends under axial compression is considered using the finite element method, and the results are validated by experimental tests. As emphasised, it is of great significance to develop a general and comprehensive solution to predict the buckling load of composite tubes and this is the main goal of the present study. This study develops an exact method, by assuming that the modal forms are to have the axial dependency in the form of simple Fourier series as formulated in Ref. 8. Stoke’s transto legitimise the formation is exploited derivatives of this Fourier series [9]. To make the stability analysis more amenable to the experimentalists, Fliigge’s shell theory [lo] has been modified for anisotropic laminated material in Ref. 11 and is used here. The method employed in the present work gives an explicit expression for exact buckling load equations. The equation is then solved for an conditions. The arbitrary set of boundary boundary conditions which are not satisfied directly by the assumed series can be enforced. The analytical method used here to determine the buckling load of multi-layered composite shells with various types of material and winding angles turns out to be general and simpler than the other available exact methods, The effect of change in geometry such as the tube length to diameter ratio, L/D and wall thickness, h, on the buckling response of isocylindrical shells anisotropic tropic and subjected to axial compression, internal and
external pressurisation are studied. We use a thin walled aluminium tube as an example of an isotropic shell, and for an anisotropic shell, a number of filament wound tubes made from glass (GRP), Kevlar (KRP) and carbon fibre reinforced plastic (CFRP). It is found that the change of L/D has a considerable influence upon the axial buckling of a shell loaded by internal pressure. In addition, the prediction of the present analysis shows good agreement with some available experimental and other theoretical results.
THEORETICAL
CONSIDERATIONS
The analytical formulation is based on Fhigge’s theory [lo]. The co-ordinate system with axes (~,t?,r) in axial, circumferential and radial directions have displacements (u,v,w), respectively. As usual the shell geometry is given by parameters L, R and h, i.e. length, radius and wall thickness, respectively. The stress-strain relations for a composite material are taken in a standard form and may be found in many standard texts on the subject [12,13]. Equations of equilibrium for elastic stability analysis The equations of elastic stability of a laminated anisotropic cylindrical shell based on Fhigge’s theory [lo] are of the form used in Ref. 13 as: A A, ,(G,> + 2 $
(GJ
A + f
(U,@)
+ (A,, +A,,)
v7X.X A + $ +-
&3 R3
D,, (VmJ - y 023
R
w~XXX --p
+ - R4 w,,,,+ W,XI)B
D,,
A *
VYXt)
~XXfl wrx
159
Buckling of composite tubes
&I
+
64,*+A33)
u 9X.X A 23 +
-
-2 -
R
%x0
033
~,w+(A~~+3
R2
7 R
’
R
v ).x.x
-2-w
)XxX-
R
R2
2K” R
“’
(2)
(WY, + v%dj) = 0
(2) and (3) the comma denotes partial differentithat subscript. The shell are given by the general
C$'[ ki,Sh’ hk-1
1, z2]dz
(4)
where C&? are the elastic moduli of the kth laminate and h,_ 1 and hk are the distances measured from the middle surface to the inner and outer surfaces of the kth laminate, respectively. The N,, &, and fix0 are the initial membrane force resultants (pre-buckling) defined as fix = K,NO, N, = K2N0 and Rx0 = K3.N0, where Ki are prescribed constants, and NO is a characteristic loading parameter.
W~XXO
A 22 -w R2
A 23 + -w,,+ wKX00 R R3
D33
-2-
+-
CD,, + 3033)
W,,H- VJ = 0
where in eqns (l), preceding a subscript ation with respect to stiffnesses A, and D, relation asl-3,14
[A,j,Dijl= D,3
(
Modal forms To analyse the buckling problem considered the displacement components are assumed to have the following form
and D23 &Xd +R”
[u,v,w] = [f,(x) +
cos n0, f2(x)
sin ne,
_w) cos nei (5) u,t)t
-
+-
A 23 R
VT, + -
A 22 R2
v,e+D,
,(w,,,,,)
Dl3 +4 R (
X CD,, + 2D33)w,,,a4
D23 - R3 (WMmJ
D22 (w,,,> + f SR”
x w-
N”
-R
f 1(x) f2(x)
mw,,,> he
where n is the circumferential mode number and the functions f,(x), f2(x) and f3(x) give the axial dependency of the various modal forms. The crucial part of the analysis involves choosing appropriate series forms for these axial mode functions. The series should be simple in form and at the same time preserve orthogonality properties. It is not necessary that the series satisfy any particular boundary conditions since we are dealing with a general case at this stage. The convenient set of Fourier series that meets all these requirements for representing the axial mode functions f,(x), f2(x), f3(x) is given as:
[f WdSl
upx-R +-R
=
3(x> I
where 13.= mdL and A,, B,, sent the Fourier coefficients.
and C,, repreEqs. can be
160
S. T S. Al-Hassani, M. Darvizeh, H. Haftchenati
designated as (CSS). It is clear from Eqs. that the Fourier sine series form always renders zero end values to displacement components v and w, hence these values can be specified as in Ref. 15
[%~vL~%wLI = [f2(O),f2(L).f3(O),f301
(7)
These end values are utilised when one uses the Stoke’s transformation to differentiate the Fourier series involved in the derivatives of relevant displacement functions. General formulation By substituting the displacement relations of (6) and their derivatives [15] into the equilibrium equations of static elastic stability (eqns e ourier coefficients are found. By (I)-(3)), th F employing the freely supported shell with no tangential constraint (FSNT) as the base problem, the general formulation outlined here is capable of dealing with buckling problems of a composite cylindrical shell with different types of loading and arbitrary end conditions. The boundary conditions for (FSNT) shell are given bY u = N,, =Q,=
g
=0
at x=O,L
(8)
In general stress resultants are not symmetric, but since thin shell structures are considered here, one can assume symmetry of stress resultants. It is observed that for this base problem of (FSNT) shell none of the eight boundary conditions are satisfied by (CSS) on a term-byterm basis. Thus, the constraint conditions are imposed to satisfy these boundary conditions [15] by the use of Stoke’s transformation. This process leads to the following problem:
[n,,l~~~,~‘l,~‘:,n;r:-,v,,,v,,w,,w,l’ =M
(9
The meanings of the symbols used above are given in the Nomenclature. For a non-trivial solution of the homogeneous eqn (9), the determinant of the coefficient matrix should necessarily be zero, i.e. det[Ljj]=O (i,j=
1,2, . ..S)
(10)
By allowing the determinant (eqn (10)) to be zero, the buckling load of (FSNT) shell is found. Solutions for shells with other boundary conditions lead to smaller size determinants.
BOUNDARY CONDITIONS Free-free As an illustration of the validity of the present analysis the results are compared with some available experimental results [16]. A circular cylindrical shell with free-free boundary conditions is considered here. This is a typical case involving the enforcement of natural boundary conditions and the release of unwanted geometrical boundary conditions. The boundary conditions of a free-free shell are given as: N,=N,o=Q,=M,=O
at x=0&
(II)
From the properties of modal forms given by eqns (5) and (6) it is clear that the tangential displacement v and radial displacement w are identically equal to zero at the shell ends x = 0, and x = L, due to the axial dependence of these parameters given in the Fourier sine series form. Therefore, the releasing procedure is required to remove these unwanted geometric boundary conditions. By defining v and w separately at the end points (eqn (7)) and including these values in the subsequent differentiation via Stoke’s transformation, the exact buckling equation is found from the enforcement of natural boundary conditions N,, = 0, QX= 0 at the ends. The characteristic equation is obtained from eqn (10) by retaining the rows and columns associated with vO,vL, wO,wL
V-J bb,vL,WO~WLIT = Ku where i, j = 5,6,7,8. With eigenvalue equation
(14 as
det[&J = 0
(13)
Clamped-clamped As another example, the results presented in Ref. 3 on anisotropic cylindrical shell and also of Ref. 7 on isotropic tubes with clampedclamped boundary conditions are compared with the present analysis. The boundary conditions for a clamped-clamped circular cylindrical shell are taken to be: 8W
u=v=w=
a,
=0 at x=O,L
(14)
161
Buckling of composite tubes
Among these conditions, conditions
the four geometric
20
i3W
u=O and a,
=0 at x=0&
(15) 15
at both ends are not automatically satisfied by the (CSS) set. Instead, this (CSS) set unnecessarily satisfies the conditions N, = 0 and M, = 0 at the ends which are not required in this case. The exact buckling equation is obtained from eqn (9) by retaining the rows and columns associated with @j!, fit, i@!, fi,L det[&J = 0 (i,j = 1,2,3,4).
H ” 8
10
% Present Method 5 --
(16)
-
Classical[ 3 ] Approximate
--
Whitney [ 3 ] 0
DISCUSSION OF RESULTS
10
20
30
40
50
60
70
80
90
Winding Angle
In order to validate the present analysis, comparison has been made with some available theoretical and experimental results [3,7,16,17]. In Ref. 3 graphite/epoxy unidirectional off-axis tubes with fixed ends were considered under axial compression loading. A comparison of buckling load to estimated strength ratio between an approximate and classical exact method was made. Using the present technique and employing an appropriate degree of convergence of the Fourier series for various winding angles good agreement is achieved. Figure 1 shows a comparison between different methods used in Ref. 3 and that of the present work. A gradual increase and eventual decrease in buckling load with increasing winding angle leads to a maximum value for a winding angle of f55” tubes. Result of buckling of elastoplastic cylindrical shells with various length to diameter ratios under axial compression are published in Ref. 7. They used aluminium alloy 6061-T6 specimens for the buckling experiments. To simulate the fixed-end boundary condition, they used a clamping set to fix the ends of the specimens with epoxy. Six strip specimens were cut from the circular cylindrical shells and the material parameters were obtained. These parameters were reported to be the Young’s modulus of elasticity E = 66.9 GPa, yield stress o,, = 240 MPa, Poisson’s ratio v = 0.34 and hardening parameter n = 2843. By simplifying the present equations an isotropic laminate theory evolves and the resulting buckling loads, by employing the above properties, are compared
Fig. 1.Comparison of buckling load for graphite/epoxy unidirectional off-axis tubes under axial compressive loading.
with those of Ref. 7 in Fig. 2. Also included in this figure are the results from the finite element analysis developed by Ref. 7. It is seen that there is a reasonable agreement between experimental, numerical and the present method. In Ref. 16, a series of tests has been carried out on a number of hoop filament wound CFRP
60 -
% ;T 40$
.
Present Method
.
Finite Element [ 7 ]
l
Experiment [ 7 ]
20 -
0 0
I
I
I
I
1
2
3
4
U-Q
Fig. 2. Comparison
of axial buckling loads for aluminium tubes for various L/D.
5
162
S. T. S. Al-Hassani, M. Darvizeh, H. Haftchenati
tubes. In these tests, the winding angles are chosen to be _+90”, f 75” and _+25”. The tubes were dynamically loaded inside a chamber which generated an external uniform compressive radial pressure sufficient to produce radial buckling. The experimental and corresponding theoretical results are shown in Table 1. Reasonable agreement was found between the experimental and the present theoretical results, except in the case of EXT 73. Only a few tests were carried out on this particular winding angle, i.e. _+75” and perhaps more tests may provide a better explanation for this difference of about 43%. It is known that, with free-free ends, long tubes subjected to internal pressure undergo column buckling. By incrementally changing tube length, an attempt was made to predict the critical buckling length of these tubes, using the present theoretical procedure. The variation of buckling internal pressure with length for tubes of various winding angle and different materials are shown in Figs 3-5. It can be seen that at lower values of L/D the buckling load asymptotically approaches very high values, well above their final internal burst pressure. These values of burst pressure have been experimentally measured for a series of short tubes L/D = 15 made of different materials with various winding angles [17] and will be discussed later in this section. As a result a critical buckling length to diameter ratio, L/D, could be identified for each particular case under consideration. By using the present method one can predict a value of L/D below which the buckling load exceeds that of corresponding burst pressure. Considering the theoretical buckling pressure for any value of L/ D, a higher value of burst pressure can be Table 1. Comparison of experimental Thickness h (mm)
Winding angle (3
Test number in Ref. 16
1.28 1.28 l-28 1.28 2.52 2.52 1.45 1.73 l-81
Hoop wound [( f 90”),],
EXT EXT EXT EXT EXT EXT EXT EXT EXT
Hoop wound [( f 90”),], K f 75”)JlS [t + 25x1,
55 56 57 59 61 62 73 84 85
expected. However, a series of experiments should be designed to investigate this phenomenon in more detail. From the above discussion it can be inferred that the higher theoretical values of buckling loads for lower L/D have no practical significance. Consequently these values have not been included and the curves are truncated as appropriate. For higher values of L/D, however, the sensitivity of the tubes to the variation of buckling load becomes less significant. This phenomenon seems to be similar for all winding angles and different materials and are shown as a plateau. Variation of the buckling load with L/D becomes steeper with increasing winding angle, which shows the sensitivity of higher winding angle tubes to buckling. From these figures it can also be seen that the onset of buckling and corresponding critical length are almost identical for different winding angles of the same material, which suggests that the critical buckling length is independent of the winding angle of the tubes. Meanwhile, the values of buckling pressure are different for various winding angles. This is to be expected due to structural dependency of composite tubes to winding angles. In fact, higher winding angle tubes are fibre dominated whereas the lower ones are matrix dominant. In general, at lower winding angles one can expect structural failure due to low values of transverse modulus which may be responsible for this kind of behaviour. This can clearly be seen in Fig. 3, where 25” CFRP tubes do not show any sign of buckling, and therefore no corresponding curve is produced, whereas KRP and GRP tubes have a tendency to fail in buckling mode. In another experimental study, 100 mm diameter thin wall composites tubes (CFRP, Kevlar, GRP) with various winding angles such as [ 161 with theoretical results Experimental buckling pressure (MPa)
Theoretical buckling pressure (MPa) 8.8
7.5 10.0 13.0 14.0 8.5 8-O 9-o
13.8 5.92 ;I:6
163
Buckling of composite tubes
. Experimental Burst Ressurej 171
10 0
O,,,,,,~,,,,,,,,,,,,I
2
4
6
8
10
12
14
16
18
20
2
4
+25”, f55” and _+75” were subjected to internal pressurisation [17]. The strain gauges were mounted in the axial direction over the gauge length along three parallel axes 120” apart. Results of these gauges revealed no effect of buckling at the lengths tested. This was confirmed by the present theoretical technique. However, further experimental results are required to verify the influence of the L/D ratio on such loading conditions. In the experimental
40 Experimental Burst Pmsure[l7]
35
z?
30
9 v
25
a & g
20
f
15
*
10
0
2
4
6
8
10
12
14
16
18
20
L/D Fig.
8
10
12
14
16
18
20
JJD
L/D
Fig. 3. Variation of critical internal pressure (to cause axial column buckling) with L/D for k 25” tubes.
l
6
4. Variation of critical internal pressure (to cause axial column buckling) with L/D for +55” tubes.
Fig. 5. Variation of critical internal pressure (to cause axial column buckling) with L/D for + 75” tubes.
procedures and analytical work, the boundary conditions are taken to be free-free.
CONCLUSION Using the present analysis it is possible to predict the critical buckling condition of thin walled composite shells, having different boundGood agreement between ary conditions. available theoretical axial compression and experimental external radial buckling pressures of composite structures and those predicted by the present analysis are obtained. In the present technique by employing the Fourier series as modal forms and truncating the series at a suitable number of terms, the results can be calculated to a desired accuracy. This, in general, leads to a more accurate analysis. The general formulation presented in this paper is capable of handling various boundary conditions with less effort and not as cumbersome as other exact available analytical methods. Although two different boundary conditions are chosen to illustrate the technique, the method can be used to implement exact applied boundary conditions. As further work it can also be employed to study the effect of geometrical and loading imperfections on the buckling behaviour of composite shell structures. It is possible to predict mode shapes and modal forces corresponding to the buckling loads for tubular composite structures with
164
many different geometries.
S. 7: S. Al-Hassani, h4. Darvizeh, H. Haftchenari
combinations
of materials
and
ACKNOWLEDGEMENTS
The authors wish to thank Dr C. B. Sharma (Department of Mathematics, UMIST) for his assistance and academic help. REFERENCES 1. Al-Hassani, S. T. S., The plastic buckling of thinwalled tubes subjected to magnetomotive forces. J. Mechanical Engineering Science 1974,16, (2) 45-70. 2. Vaughan, H. and Florance, A. L., Plastic flow buckling of cylindrical shells due to impulsive loading. Trans. ASME J. Appl. Mech., 1970, 171-9. 3. Whitney, J. M. and Sun, C. T., Buckling of composite cylindrical characterisation specimens. J. Comp. Mat. 1975,9, 138-148. 4. Tse, P., Failure of filament wound glass reinforced plastic tubes under complex loading. Ph.D. Thesis, Mech. Eng. Department, UMIST, 1983. 5. Hu, H. T., Influence of in-plane shear non-linearity on buckling and postbuckling response of composite
$a8tzi5and
shells.
J.
Comp. Mat. 1993,
27,
(2)
6. Yan, F. G., Composite laminated shells under internal pressure. AIAA Journal (Technical Notes) 1992, 30, (6), 1669-1672. 7. Lin, M. C. and Yeh, M. K., Buckling of elastoplastic circular cylindrical shells under axial compression. AJAA Journal 1994,32, (11) 2309-2315. 8. Sharma, C. B., Darvizeh, M. and Darvizeh, A., Free vibration response of multilayered orthotropic fluidfilled circular cylindrical shells. Journal of Comp. Structures, 1996, 34, 349-355. 9. Darvizeh, M. and Sharma, C. B., Free vibration behaviour of helically wound tubular structures. Journal of Composites, Part B: Engineering (in press). 10. Fliigge, W., Stresses in Shells. Springer, Berlin, 1962. 11. Cheng, S. and Ho, B. P. C., Stability of heterogeneous aelotropic cylindrical shells under combined loading. AJAA Journal 1963, 1,892-905. 12. Jones, R. M., Mechanics of Composite Materials. Scripta Book Company, Washington, D.C., 1975. 13. Datoo, M. H., Mechanics Of Fibrous Composites. Elsevier Applied Science, UK, 1991. 14. Darvizeh, M. and Sharma, C. B., Natural frequencies of laminated orthotropic thin circular cylinders. Int. J. Thin Walled Struct. 1984,2, 207-217. 15. Darvizeh, M., Free vibration characteristics of orthotropic thin circular cylindrical shells. Ph.D. Thesis, University of Manchester, 1986. 16. Ahmad, R., Ph.D. Thesis, UMIST, 1996. 17. Haftchenari, H., Ph.D. Thesis, UMIST, 1997.
PII:SO263-8223(97)00135-9
An analytical study of buckling of composite tubes with various boundary conditions S. T. S. Al-Hassani, M. Darvizeh* & H. Haftchenari Applied Mechanics Division, Mechanical Engineering Dept., UMIST PO Box 88, Sackville Street, Manchester M60 lQD, UK
A theoretical investigation of the buckling problem of composite tubes with different boundary conditions is presented. Different types of loading such as compressive, torsional, internal, external or any combination of them can be applied. To increase the accuracy of results, the modal forms are assumed to have axial dependency in the form of simple Fourier series. The derivatives of the Fourier series are legitimised using Stoke’s transformation. Fhigge’s shell equations, modified for anisotropic laminated materials, are used. The method developed in this work for finding the buckling loads yields an exact equation which is simpler than the exact method adopted by other research workers. To check the validity of the present analysis, the results are compared with some available theoretical and experimental results. 0 1997 Elsevier Science Ltd.
NOMENCLATURE
INTRODUCTION
The buckling of fibrous composite structures with different boundary conditions and various types of loading is of particular interest to engineers in modern technology. This is due to the importance of basic load carrying members of advanced structures such as aircraft fuselages, submarine hulls, booster rockets, pipelines and pressure vessels. The introduction of high strength to weight ratio of fibre-reinforced composite material has led to greater flexibility in the design of advanced structures. As a result there has been a considerable interest in developing a viable testing technique to characterise the mechanical properties of composites. It has been realised that testing of tubular specimens may lead to more reliable results and some problems such as end conditions and other implications in testing procedures are less significant compared with tests on coupon type specimens where end gripping exists. A typical problem with testing tubular specimens is the presence of buckling. Designers and experimentalists are often interested in a quick and accurate estimate of the buckling load which normally requires a comprehensive development of an exact mathematical model.
elastic moduli of the kth laminate shell stiffnesses Fourier coefficients shell diameter transverse modulus matrix elements thickness of the kth shell layer total shell thickness shell length axial mode number circumferential mode number shell radius N,(O, B)/cos ntl -N,(L, B)/cos ne - 7cbiyo,ql(L cos pq
nM,(L,ey(~ cos ne) in-plane and radial displacement components axial and circumferential coordinates
*On leave from the Department of Mechanical Engineering, University of Guilan, PO Box 401, Rasht, Iran.
157
158
S. 7: S. Al-Hassani, M. Darvizeh, H. Haftchenati
In Ref. 1 the plastic buckling of thin-walled aluminium tubes and rings of different sizes, under intense external magnetic pressure has been investigated. The equations related to the radial buckling analysis given by Ref. 2 have also been developed. In Ref. 3, theoretical results using exact and approximate methods are presented for shells under axial compression and torsion with clamped-clamped boundary conditions. Experimental results concerning the failure prediction of filament wound glass reinforced plastics are presented in Ref. 4. In Ref. 5, the finite element method is used to evaluate the effects of in-plane shear non-linearity on buckling response of clamped-clamped shells. A theoretical study of the response of filament wound composite shells with clamped ends under internal pressure is presented in Ref. 6. In Ref. 7, the buckling of elastoplastic circular cylindrical shells with clamped ends under axial compression is considered using the finite element method, and the results are validated by experimental tests. As emphasised, it is of great significance to develop a general and comprehensive solution to predict the buckling load of composite tubes and this is the main goal of the present study. This study develops an exact method, by assuming that the modal forms are to have the axial dependency in the form of simple Fourier series as formulated in Ref. 8. Stoke’s transto legitimise the formation is exploited derivatives of this Fourier series [9]. To make the stability analysis more amenable to the experimentalists, Fliigge’s shell theory [lo] has been modified for anisotropic laminated material in Ref. 11 and is used here. The method employed in the present work gives an explicit expression for exact buckling load equations. The equation is then solved for an conditions. The arbitrary set of boundary boundary conditions which are not satisfied directly by the assumed series can be enforced. The analytical method used here to determine the buckling load of multi-layered composite shells with various types of material and winding angles turns out to be general and simpler than the other available exact methods, The effect of change in geometry such as the tube length to diameter ratio, L/D and wall thickness, h, on the buckling response of isocylindrical shells anisotropic tropic and subjected to axial compression, internal and
external pressurisation are studied. We use a thin walled aluminium tube as an example of an isotropic shell, and for an anisotropic shell, a number of filament wound tubes made from glass (GRP), Kevlar (KRP) and carbon fibre reinforced plastic (CFRP). It is found that the change of L/D has a considerable influence upon the axial buckling of a shell loaded by internal pressure. In addition, the prediction of the present analysis shows good agreement with some available experimental and other theoretical results.
THEORETICAL
CONSIDERATIONS
The analytical formulation is based on Fhigge’s theory [lo]. The co-ordinate system with axes (~,t?,r) in axial, circumferential and radial directions have displacements (u,v,w), respectively. As usual the shell geometry is given by parameters L, R and h, i.e. length, radius and wall thickness, respectively. The stress-strain relations for a composite material are taken in a standard form and may be found in many standard texts on the subject [12,13]. Equations of equilibrium for elastic stability analysis The equations of elastic stability of a laminated anisotropic cylindrical shell based on Fhigge’s theory [lo] are of the form used in Ref. 13 as: A A, ,(G,> + 2 $
(GJ
A + f
(U,@)
+ (A,, +A,,)
v7X.X A + $ +-
&3 R3
D,, (VmJ - y 023
R
w~XXX --p
+ - R4 w,,,,+ W,XI)B
D,,
A *
VYXt)
~XXfl wrx
159
Buckling of composite tubes
&I
+
64,*+A33)
u 9X.X A 23 +
-
-2 -
R
%x0
033
~,w+(A~~+3
R2
7 R
’
R
v ).x.x
-2-w
)XxX-
R
R2
2K” R
“’
(2)
(WY, + v%dj) = 0
(2) and (3) the comma denotes partial differentithat subscript. The shell are given by the general
C$'[ ki,Sh’ hk-1
1, z2]dz
(4)
where C&? are the elastic moduli of the kth laminate and h,_ 1 and hk are the distances measured from the middle surface to the inner and outer surfaces of the kth laminate, respectively. The N,, &, and fix0 are the initial membrane force resultants (pre-buckling) defined as fix = K,NO, N, = K2N0 and Rx0 = K3.N0, where Ki are prescribed constants, and NO is a characteristic loading parameter.
W~XXO
A 22 -w R2
A 23 + -w,,+ wKX00 R R3
D33
-2-
+-
CD,, + 3033)
W,,H- VJ = 0
where in eqns (l), preceding a subscript ation with respect to stiffnesses A, and D, relation asl-3,14
[A,j,Dijl= D,3
(
Modal forms To analyse the buckling problem considered the displacement components are assumed to have the following form
and D23 &Xd +R”
[u,v,w] = [f,(x) +
cos n0, f2(x)
sin ne,
_w) cos nei (5) u,t)t
-
+-
A 23 R
VT, + -
A 22 R2
v,e+D,
,(w,,,,,)
Dl3 +4 R (
X CD,, + 2D33)w,,,a4
D23 - R3 (WMmJ
D22 (w,,,> + f SR”
x w-
N”
-R
f 1(x) f2(x)
mw,,,> he
where n is the circumferential mode number and the functions f,(x), f2(x) and f3(x) give the axial dependency of the various modal forms. The crucial part of the analysis involves choosing appropriate series forms for these axial mode functions. The series should be simple in form and at the same time preserve orthogonality properties. It is not necessary that the series satisfy any particular boundary conditions since we are dealing with a general case at this stage. The convenient set of Fourier series that meets all these requirements for representing the axial mode functions f,(x), f2(x), f3(x) is given as:
[f WdSl
upx-R +-R
=
3(x> I
where 13.= mdL and A,, B,, sent the Fourier coefficients.
and C,, repreEqs. can be
160
S. T S. Al-Hassani, M. Darvizeh, H. Haftchenati
designated as (CSS). It is clear from Eqs. that the Fourier sine series form always renders zero end values to displacement components v and w, hence these values can be specified as in Ref. 15
[%~vL~%wLI = [f2(O),f2(L).f3(O),f301
(7)
These end values are utilised when one uses the Stoke’s transformation to differentiate the Fourier series involved in the derivatives of relevant displacement functions. General formulation By substituting the displacement relations of (6) and their derivatives [15] into the equilibrium equations of static elastic stability (eqns e ourier coefficients are found. By (I)-(3)), th F employing the freely supported shell with no tangential constraint (FSNT) as the base problem, the general formulation outlined here is capable of dealing with buckling problems of a composite cylindrical shell with different types of loading and arbitrary end conditions. The boundary conditions for (FSNT) shell are given bY u = N,, =Q,=
g
=0
at x=O,L
(8)
In general stress resultants are not symmetric, but since thin shell structures are considered here, one can assume symmetry of stress resultants. It is observed that for this base problem of (FSNT) shell none of the eight boundary conditions are satisfied by (CSS) on a term-byterm basis. Thus, the constraint conditions are imposed to satisfy these boundary conditions [15] by the use of Stoke’s transformation. This process leads to the following problem:
[n,,l~~~,~‘l,~‘:,n;r:-,v,,,v,,w,,w,l’ =M
(9
The meanings of the symbols used above are given in the Nomenclature. For a non-trivial solution of the homogeneous eqn (9), the determinant of the coefficient matrix should necessarily be zero, i.e. det[Ljj]=O (i,j=
1,2, . ..S)
(10)
By allowing the determinant (eqn (10)) to be zero, the buckling load of (FSNT) shell is found. Solutions for shells with other boundary conditions lead to smaller size determinants.
BOUNDARY CONDITIONS Free-free As an illustration of the validity of the present analysis the results are compared with some available experimental results [16]. A circular cylindrical shell with free-free boundary conditions is considered here. This is a typical case involving the enforcement of natural boundary conditions and the release of unwanted geometrical boundary conditions. The boundary conditions of a free-free shell are given as: N,=N,o=Q,=M,=O
at x=0&
(II)
From the properties of modal forms given by eqns (5) and (6) it is clear that the tangential displacement v and radial displacement w are identically equal to zero at the shell ends x = 0, and x = L, due to the axial dependence of these parameters given in the Fourier sine series form. Therefore, the releasing procedure is required to remove these unwanted geometric boundary conditions. By defining v and w separately at the end points (eqn (7)) and including these values in the subsequent differentiation via Stoke’s transformation, the exact buckling equation is found from the enforcement of natural boundary conditions N,, = 0, QX= 0 at the ends. The characteristic equation is obtained from eqn (10) by retaining the rows and columns associated with vO,vL, wO,wL
V-J bb,vL,WO~WLIT = Ku where i, j = 5,6,7,8. With eigenvalue equation
(14 as
det[&J = 0
(13)
Clamped-clamped As another example, the results presented in Ref. 3 on anisotropic cylindrical shell and also of Ref. 7 on isotropic tubes with clampedclamped boundary conditions are compared with the present analysis. The boundary conditions for a clamped-clamped circular cylindrical shell are taken to be: 8W
u=v=w=
a,
=0 at x=O,L
(14)
161
Buckling of composite tubes
Among these conditions, conditions
the four geometric
20
i3W
u=O and a,
=0 at x=0&
(15) 15
at both ends are not automatically satisfied by the (CSS) set. Instead, this (CSS) set unnecessarily satisfies the conditions N, = 0 and M, = 0 at the ends which are not required in this case. The exact buckling equation is obtained from eqn (9) by retaining the rows and columns associated with @j!, fit, i@!, fi,L det[&J = 0 (i,j = 1,2,3,4).
H ” 8
10
% Present Method 5 --
(16)
-
Classical[ 3 ] Approximate
--
Whitney [ 3 ] 0
DISCUSSION OF RESULTS
10
20
30
40
50
60
70
80
90
Winding Angle
In order to validate the present analysis, comparison has been made with some available theoretical and experimental results [3,7,16,17]. In Ref. 3 graphite/epoxy unidirectional off-axis tubes with fixed ends were considered under axial compression loading. A comparison of buckling load to estimated strength ratio between an approximate and classical exact method was made. Using the present technique and employing an appropriate degree of convergence of the Fourier series for various winding angles good agreement is achieved. Figure 1 shows a comparison between different methods used in Ref. 3 and that of the present work. A gradual increase and eventual decrease in buckling load with increasing winding angle leads to a maximum value for a winding angle of f55” tubes. Result of buckling of elastoplastic cylindrical shells with various length to diameter ratios under axial compression are published in Ref. 7. They used aluminium alloy 6061-T6 specimens for the buckling experiments. To simulate the fixed-end boundary condition, they used a clamping set to fix the ends of the specimens with epoxy. Six strip specimens were cut from the circular cylindrical shells and the material parameters were obtained. These parameters were reported to be the Young’s modulus of elasticity E = 66.9 GPa, yield stress o,, = 240 MPa, Poisson’s ratio v = 0.34 and hardening parameter n = 2843. By simplifying the present equations an isotropic laminate theory evolves and the resulting buckling loads, by employing the above properties, are compared
Fig. 1.Comparison of buckling load for graphite/epoxy unidirectional off-axis tubes under axial compressive loading.
with those of Ref. 7 in Fig. 2. Also included in this figure are the results from the finite element analysis developed by Ref. 7. It is seen that there is a reasonable agreement between experimental, numerical and the present method. In Ref. 16, a series of tests has been carried out on a number of hoop filament wound CFRP
60 -
% ;T 40$
.
Present Method
.
Finite Element [ 7 ]
l
Experiment [ 7 ]
20 -
0 0
I
I
I
I
1
2
3
4
U-Q
Fig. 2. Comparison
of axial buckling loads for aluminium tubes for various L/D.
5
162
S. T. S. Al-Hassani, M. Darvizeh, H. Haftchenati
tubes. In these tests, the winding angles are chosen to be _+90”, f 75” and _+25”. The tubes were dynamically loaded inside a chamber which generated an external uniform compressive radial pressure sufficient to produce radial buckling. The experimental and corresponding theoretical results are shown in Table 1. Reasonable agreement was found between the experimental and the present theoretical results, except in the case of EXT 73. Only a few tests were carried out on this particular winding angle, i.e. _+75” and perhaps more tests may provide a better explanation for this difference of about 43%. It is known that, with free-free ends, long tubes subjected to internal pressure undergo column buckling. By incrementally changing tube length, an attempt was made to predict the critical buckling length of these tubes, using the present theoretical procedure. The variation of buckling internal pressure with length for tubes of various winding angle and different materials are shown in Figs 3-5. It can be seen that at lower values of L/D the buckling load asymptotically approaches very high values, well above their final internal burst pressure. These values of burst pressure have been experimentally measured for a series of short tubes L/D = 15 made of different materials with various winding angles [17] and will be discussed later in this section. As a result a critical buckling length to diameter ratio, L/D, could be identified for each particular case under consideration. By using the present method one can predict a value of L/D below which the buckling load exceeds that of corresponding burst pressure. Considering the theoretical buckling pressure for any value of L/ D, a higher value of burst pressure can be Table 1. Comparison of experimental Thickness h (mm)
Winding angle (3
Test number in Ref. 16
1.28 1.28 l-28 1.28 2.52 2.52 1.45 1.73 l-81
Hoop wound [( f 90”),],
EXT EXT EXT EXT EXT EXT EXT EXT EXT
Hoop wound [( f 90”),], K f 75”)JlS [t + 25x1,
55 56 57 59 61 62 73 84 85
expected. However, a series of experiments should be designed to investigate this phenomenon in more detail. From the above discussion it can be inferred that the higher theoretical values of buckling loads for lower L/D have no practical significance. Consequently these values have not been included and the curves are truncated as appropriate. For higher values of L/D, however, the sensitivity of the tubes to the variation of buckling load becomes less significant. This phenomenon seems to be similar for all winding angles and different materials and are shown as a plateau. Variation of the buckling load with L/D becomes steeper with increasing winding angle, which shows the sensitivity of higher winding angle tubes to buckling. From these figures it can also be seen that the onset of buckling and corresponding critical length are almost identical for different winding angles of the same material, which suggests that the critical buckling length is independent of the winding angle of the tubes. Meanwhile, the values of buckling pressure are different for various winding angles. This is to be expected due to structural dependency of composite tubes to winding angles. In fact, higher winding angle tubes are fibre dominated whereas the lower ones are matrix dominant. In general, at lower winding angles one can expect structural failure due to low values of transverse modulus which may be responsible for this kind of behaviour. This can clearly be seen in Fig. 3, where 25” CFRP tubes do not show any sign of buckling, and therefore no corresponding curve is produced, whereas KRP and GRP tubes have a tendency to fail in buckling mode. In another experimental study, 100 mm diameter thin wall composites tubes (CFRP, Kevlar, GRP) with various winding angles such as [ 161 with theoretical results Experimental buckling pressure (MPa)
Theoretical buckling pressure (MPa) 8.8
7.5 10.0 13.0 14.0 8.5 8-O 9-o
13.8 5.92 ;I:6
163
Buckling of composite tubes
. Experimental Burst Ressurej 171
10 0
O,,,,,,~,,,,,,,,,,,,I
2
4
6
8
10
12
14
16
18
20
2
4
+25”, f55” and _+75” were subjected to internal pressurisation [17]. The strain gauges were mounted in the axial direction over the gauge length along three parallel axes 120” apart. Results of these gauges revealed no effect of buckling at the lengths tested. This was confirmed by the present theoretical technique. However, further experimental results are required to verify the influence of the L/D ratio on such loading conditions. In the experimental
40 Experimental Burst Pmsure[l7]
35
z?
30
9 v
25
a & g
20
f
15
*
10
0
2
4
6
8
10
12
14
16
18
20
L/D Fig.
8
10
12
14
16
18
20
JJD
L/D
Fig. 3. Variation of critical internal pressure (to cause axial column buckling) with L/D for k 25” tubes.
l
6
4. Variation of critical internal pressure (to cause axial column buckling) with L/D for +55” tubes.
Fig. 5. Variation of critical internal pressure (to cause axial column buckling) with L/D for + 75” tubes.
procedures and analytical work, the boundary conditions are taken to be free-free.
CONCLUSION Using the present analysis it is possible to predict the critical buckling condition of thin walled composite shells, having different boundGood agreement between ary conditions. available theoretical axial compression and experimental external radial buckling pressures of composite structures and those predicted by the present analysis are obtained. In the present technique by employing the Fourier series as modal forms and truncating the series at a suitable number of terms, the results can be calculated to a desired accuracy. This, in general, leads to a more accurate analysis. The general formulation presented in this paper is capable of handling various boundary conditions with less effort and not as cumbersome as other exact available analytical methods. Although two different boundary conditions are chosen to illustrate the technique, the method can be used to implement exact applied boundary conditions. As further work it can also be employed to study the effect of geometrical and loading imperfections on the buckling behaviour of composite shell structures. It is possible to predict mode shapes and modal forces corresponding to the buckling loads for tubular composite structures with
164
many different geometries.
S. 7: S. Al-Hassani, h4. Darvizeh, H. Haftchenari
combinations
of materials
and
ACKNOWLEDGEMENTS
The authors wish to thank Dr C. B. Sharma (Department of Mathematics, UMIST) for his assistance and academic help. REFERENCES 1. Al-Hassani, S. T. S., The plastic buckling of thinwalled tubes subjected to magnetomotive forces. J. Mechanical Engineering Science 1974,16, (2) 45-70. 2. Vaughan, H. and Florance, A. L., Plastic flow buckling of cylindrical shells due to impulsive loading. Trans. ASME J. Appl. Mech., 1970, 171-9. 3. Whitney, J. M. and Sun, C. T., Buckling of composite cylindrical characterisation specimens. J. Comp. Mat. 1975,9, 138-148. 4. Tse, P., Failure of filament wound glass reinforced plastic tubes under complex loading. Ph.D. Thesis, Mech. Eng. Department, UMIST, 1983. 5. Hu, H. T., Influence of in-plane shear non-linearity on buckling and postbuckling response of composite
$a8tzi5and
shells.
J.
Comp. Mat. 1993,
27,
(2)
6. Yan, F. G., Composite laminated shells under internal pressure. AIAA Journal (Technical Notes) 1992, 30, (6), 1669-1672. 7. Lin, M. C. and Yeh, M. K., Buckling of elastoplastic circular cylindrical shells under axial compression. AJAA Journal 1994,32, (11) 2309-2315. 8. Sharma, C. B., Darvizeh, M. and Darvizeh, A., Free vibration response of multilayered orthotropic fluidfilled circular cylindrical shells. Journal of Comp. Structures, 1996, 34, 349-355. 9. Darvizeh, M. and Sharma, C. B., Free vibration behaviour of helically wound tubular structures. Journal of Composites, Part B: Engineering (in press). 10. Fliigge, W., Stresses in Shells. Springer, Berlin, 1962. 11. Cheng, S. and Ho, B. P. C., Stability of heterogeneous aelotropic cylindrical shells under combined loading. AJAA Journal 1963, 1,892-905. 12. Jones, R. M., Mechanics of Composite Materials. Scripta Book Company, Washington, D.C., 1975. 13. Datoo, M. H., Mechanics Of Fibrous Composites. Elsevier Applied Science, UK, 1991. 14. Darvizeh, M. and Sharma, C. B., Natural frequencies of laminated orthotropic thin circular cylinders. Int. J. Thin Walled Struct. 1984,2, 207-217. 15. Darvizeh, M., Free vibration characteristics of orthotropic thin circular cylindrical shells. Ph.D. Thesis, University of Manchester, 1986. 16. Ahmad, R., Ph.D. Thesis, UMIST, 1996. 17. Haftchenari, H., Ph.D. Thesis, UMIST, 1997.