On the buckling and collapse of moderately thick composite cylinders under hydrostatic pressure

ELSEVIER PIh S1359-8368(96)00072-8

Composites Part B 28B (1997) 583-596 © 1997 Elsevier Science Limited

Printed in Great Britain. All rights reserved 1359-8368/97/$17.00

On the buckling and collapse of moderately thick composite cylinders under hydrostatic pressure

Srinivasan Sridharan a and Akihito Kasagi b aWashington University, Campus Box 1130, St. Louis, MO 63130, USA bKozo Keikaku Engineering Inc., Tokyo, Japan (Received 8 May 1995; accepted 15 October 1996)

This paper presents a summary of the work carried out at Washington University in recent years on the buckling and associated non-linear response and collapse of moderately thick composite cylindrical shells. Ring elements in conjunction with a three-dimensional elasticity formulation are employed in the analysis. The buckling and posthuckling imperfection sensitivity in individual modes is studied first. The problem of interaction between local and overall instabilities is then investigated in detail. Imperfection sensitivity of typical ring-stiffened shells is established by using a simple and effective approach that combines the asymptotic procedure and the amplitude- modulation technique. The influence of dynamic application of the hydrostatic pressure is investigated with the simplified model. The results obtained are compared with those produced by a two-dimensional program package which includes full-fledged non-linear analysis with ring elements, and commercial programs wherever possible. The study has thrown light on several issues regarding the modeling and behavioral aspects of thick composite shells which are summarized at the conclusion of the paper. © 1997 Elsevier Science Limited.

(Keywords: B. buckling; C. analytical modelling; composite cylinders)

INTRODUCTION

The rapidly increasing use of high-performance composite materials in the fabrication of structural components and systems has provided the motivation for vigorous and sustained research in this field for the past several decades. In recent times there has been considerable interest in the structural instability of relatively thick composite shells; however, attention in the literature has so far been confined to the initial buckling problems 1-3. A search for an optimal design of composite cylindrical shells, of the type proposed to be used as submarine vessels, leads to a geometry for which buckling and the accompanying non-linear behavior are of major concern. These shells are usually, in the interests of economy and lightness, reinforced best by ring stiffeners to resist severe hydrostatic pressure. Such stiffened shells can suffer buckling not only in the overall sense, but also locally between the stiffeners. These two distinctive buckling modes, which dominate the behavior of ring stiffened shells, are described as follows:

(1) The short-wave local buckling in which buckling occurs between stiffeners, forming several half waves in the circumferential direction. The nodes of buckling waves in the longitudinal direction coincide with the position of stiffeners so that the role of stiffeners may be viewed as those of supports which arrest the

movement of the shell in the radial direction. However, this involves bending action of the stiffeners as a result of buckling.

(2) The long-wave overall buckling where the stiffeners are pulled radially in and out; the mode would consist of a relatively small number of waves in the circumferential direction.

It is well known that the adverse non-linear interaction between these modes is the principal cause of failure of optimally designed shell structures for which the critical stresses corresponding to these modes would be close. Thus, it is essential to understand the nature of the response, such as possible posthuckling resistance or imperfection sensitivity and the interaction of overall and local instabilities.

The present study deals with the aforementioned issues of buckling of moderately thick composite shells under hydrostatic pressure. A three-dimensional (3D) formulation is pursued to capture the shear deformation effects and to model precisely the structural action of the junctions of stiffeners with the shell. Much emphasis has been placed on capturing the essential features of the structural response with utmost computational economy. To this end, an asymptotic procedure has been used to investigate the postbuckling response and imperfection sensitivity in individual modes. The information thus obtained is utilized

583

Buckling and collapse of cylinders under hydrostatic pressure: S. Sridharan and A. Kasagi

(a)

h

")J

qo

r

/ f

~ f - . "-'~-z- J

0 ~ " ~ q0

(b)

r , w l' o

(c)

L I

I h R 0

I

t

S ~ts ,1 P I ts_l

' T I

I Shell skin II Junction III Stiffener

Figure 1 (a) Shell geometry, loading and the coordinate system; (b) longitudinal section of the shell; and (c) key dimensions and finite element configuration

in the analysis of modal interaction. It has been shown 4 that, as the local mode comes under the influence of the overall mode, the local buckling deflections are accentuated in parts of the shell which buckle inwards due to overall buckling. This effect is accounted for in the present model by associating a 'modulating function' with the local buckling deformations 5. Thus a computationally inexpensive model is developed to investigate the related issues of imperfection

sensitivity and optimality. It is well known that imperfection- sensitive structures suffer greater loss of load-carrying capacity under suddenly applied loading compared with quasi-static loading. This phenomenon is briefly explored in the context of ring-stiffened shells in the present study. The paper concludes with comparisons of the numerical results for buckling and non-linear response with those obtained by using a two-dimensional (2D) formulation and currently available commercial codes.

METHODOLOGY

General details o f the formulation

The geometry, loading and the coordinate system of the shell are shown in Figure l(a). A longitudinal section of the shell with stiffeners is shown in Figure l(b).

With a view to having the capability to treat thick shells, the three-dimensional constitutive relationships and strain- displacement relationships are used throughout. These are briefly introduced in the sequel. The features of the finite element formulation common to all stages of the analysis, i.e. the shape functions, the arrangement of the elements and the numerical integration scheme, are discussed next.

Constitutive relationships. Consider a curved lamina whose principal material directions are defined by the coordinate system 1-2-3, where axis 3 coincides with the outward normal at any point, axis 1 coincides with the fiber direction and the axes 1-2 are obtained by rotating the tangents in the longitudinal and radial directions through an angle or. The reference coordinate system at any point is defined by longitudinal (x-axis), circumferential (0-axis) and radial (r-axis) coordinates, as shown in Figure l(a), and the corresponding displacement components are respectively u, v and w. Simi- larly, the material principal axes of a ring stiffener are defined by the angle between the fiber and radial directions. The relationship between stresses and strains with reference to the global coordinate system takes the form:

~Cll

C12

C13

0

0

Cl6

C12 Cl3 0 0

C22 C23 0 0

C23 6"33 0 0

0 0 C44 C45

0 0 C45 C55

C26 C36 0 0

' Ol '

O" 2

173

a4

05

, G 6 ,

Subscripts 1-6 are defined as follows:

For the shell:

1---*xx, 2---,00, 3 ---~ rr, 4 - * rO,

C16 /31

C26 /32

C36 /33

0 /34

0 /35

C66 /36

( la-f)

For the stiffener:

1 ---, rr, 2---* 00, 3 ~ xx, 4--* Ox,

5 --~ rx, 6---. Ox

(2a)

5 --* rx, 6 ---, rO

(2b)

584


The coefficients of eqn (1) are obtainable at any location by a transformation to the axes of the structure of the elastic constants referred to the principal material axes 6. Further, in order to improve the numerical conditioning, we set, for stiffeners, trx~ (the normal stress in the thickness direction) equal to 0. This leads to the reduced stiffness, Cij, for the stiffener material:

Cij ---- Cij -- C3--~3C3j Ci3 (i,j = 1,2, 6) Cij = C o (i,j = 4, 5) (3)

The subscript 3 for the stiffener is absent throughout in the present treatment.

Strain-displacement relationships. Polar coordinates, x, 0 and r [Figure l(a)], are used in expressing the relationships between the strain components and the displacement components, u, v and w. These are taken in the most general forms given by Ambartsumyan 7 and Kasagi and Sridharan 4 and will not be repeated here. Strains are considered to be infinitesimally small, but displacements and their gradients can be large.

Loading and boundary conditions. The hydrostatic pressure loading may be viewed as being made up of two parts: external pressure, qo, and axial force, P, communicated to the shell from the closed ends as shown in Figure l(a). The shells are assumed to be simply supported at their ends, v = w = 0. Pressure is either radially directed (dead load) or always normal to the shell surface (live load). This subject is discussed in greater detail elsewhere.

Finite element formulation. Figure l(b) shows the longitudinal section of the shell. The displacement functions employed are polynomials in the longitudinal ( -x) and radial ( - r ) directions and trigonometric functions in the circumferential direction. The appropriate forms of the trigonometric variations will be discussed in the later sections. The polynomials employed form a set of hierarchical functions and are obtained through a process of integration of Legendre polynomials 8. These are given explicitly in earlier papers of the authors 9'4. Figure l(c) shows the key dimensions of a typical bay and the discretization scheme. There are three classes of element, those belonging respectively to the shell skin (I), the stiffener (II) and the junction (III). The polynomial levels (p-levels), Px and Pr, chosen for these elements must be mutually consistent so that complete displacement compatibility is ensured at the common bound- aries, as shown in Table 1. Numerical convergence of the results is checked by keeping the element configuration unchanged and simply increasing the p-levels.

A typical element is so formulated that it can consist of several layers. Because the material properties vary within the element, the limits of integration and the constitutive relationships of say the kth layer (Figure 2) must be defined before numerical integration is carried out.

Prebuckling analysis

Because of the variation of the prebuckling stresses in the

Table 1 Selection o f po lynomia l levels

Part Descr ipt ion p- levels

r-direct ion x-direct ion

I Skin Pr px II Junct ion P r P r III Stiffener P x P r

longitudinal direction, particularly in the case of stiffened shells, it is necessary to evaluate the prebuckling stress state. Both the stress distribution and deformations are axisymmetric for a perfect shell. The prebuckling state is computed from linearized strain-displacement relationships. The displacement fields are taken in the form:

U ( 0 ) - - - u O d p i ( ~ ) ~ j ( ~ )

v (°) = v°4~i(~)daj(~) (i = l, 2 . . . . . Px + l; j = 1,2 ..... Pr + l)

w (o) = w ° ~ ( ~ ) ~ j ( O

(4a--c)

where u °, v ° and w ° are the degrees-of-freedom and (~i(~') and ~bj(~t) are shape functions, P r and Px stand for polynomial levels that are independently selected for radial and longitudinal directions, respectively.

Buckling analysis

The buckling analysis is performed by assuming the prebuckling state to be linear, i.e. only the corresponding stress distribution is taken into account and the deflections

R]

R 0

/th element

OuterLe surface T [ i m ' ' ~ ~ - - ' ~ I h

,-~ =--I IRr_ Re/

Inner surface

t+l / k th layer

""i~" ,::" ~-~- "" ~ j ~ 1 ~ t(k)

± N ~ fth ELEMENT

Figure 2 Numerica l integrat ion scheme

]I~X

585


are simply neglected. This so-called linear stability analysis is sometimes called into question and rightly so for thin shells, but for relatively thick shells it appears to be justified. A discussion on this is postponed until later in the paper.

The displacement functions characterizing the buckling mode are expressed in terms of the degrees-of-freedom, ubS~ U IC, i etc., and 4K~'), 4ff~):

u (l) = u~jS4~(~)4)j(~l) sin(n0) ÷ ubCrbi(~)4~j(~) cos(n0)

(5a--c)

v O) = vlS4)i(f)4~j(~) sin(n0) ÷ v~C4~i(f)4)j(~) cos(n0)

w (1) = w~S4~i(f)4)j(~l) sin(n0) + w~.C4)i(f)4~(~) cos(n0)

( i = 1,2 . . . . . p r ÷ l ; j = l , 2 . . . . . p x ÷ l )

where Pr and Px represent p-levels for the radial and longitudinal directions, respectively.

A potential energy function for the linear stability problem is invoked (see, for example, 9,4). This is a homogeneous quadratic and leads to the linear eigenvalue problem of the general type:

(1) ~,.(1)'~ (1) aij -^o i j )q) =0 ( i , j = l , 2 . . . . . Ul) (6)

in which N1 is the total number of degrees-of-freedom of the first-order problem, and X is the load parameter (in the present context = q0). For an anisotropic shell there occurs a coupling of all the degrees-of-freedom which will be designated generically as q(l).

Fluid pressure loading

In order to describe the loading condition precisely as the shell buckles, it is necessary to consider a 'live fluid pressure' loading condition where the pressure at any instant remains normal to the deformed surface. Brush and Almroth 1° and Koiter ll have discussed in detail the problems of the fluid pressure loading, showing that the loading is still conservative and a potential function of the external load exists. This function can be obtained by multiplying the pressure by the change in the volume enclosed by the shell as the deformation occurs. In general, the expression for the loss of potential due to the fluid pressure condition consists of linear and quadratic terms. However, in setting up the total potential energy function, all the linear terms vanish in the buckled state and only quadratic terms remain. Thus the potential energy contribu- tion from fluid pressure is taken in the simplified form given bye°:

~ = ~q0~; Iv 2 v(OW) (0v ] where the integration is taken over the outer surface of the shell. This term must be included in the potential energy of the buckled state. Its effect is especially important in the analysis of overall buckling with n < 4.

Asymptotic procedure

In the study of postbuckling behavior, we adopt the

asymptotic approach developed by Koiter 12 and discussed by Budiansky and Hutchinson 13. The displacement { u }, the strain {E} and the stress {a} vectors associated with buckling are each taken in the form of a power series in a scalar parameter,

[n*} =/U(1)}~ ÷/U(2)}~ 2 ÷ {U(3)}~j 3 ÷ ' - -

{e*l = [80)1~ +/e(2)/~ 2 + [e(3)}~ 3 + " " (8a-c)

{a*} = [tr(1)}~ ÷ [o-(2)}~ 2 ÷ {o(3)}~ 3 ÷ . . .

Each of {u")}, {E (1)} and {a (1)} (first-order quantities) are given by the buckling mode while the terms {u(2)}, {c (2)} and {o 12)} are the respective second-order fields, and so on.

measures the growth of the deformation in the form of the mode of buckling in the postbuckled state whereas the higher order fields provide a modification of the deformation pattern. The present study considers up to the second- order field to predict the postbuckling response and evaluate the stress distribution in the vicinity of bifurcation.

Second-order field problem

A potential energy function governing the second-order field problem is invoked 9 which, when rendered stationary, provides the equations goveming the field. The displacement functions describing the second-order field are assumed in the form:

20 2S U (2) = Uij ~i(~)~j(l"]) ÷ Uij ~i(~)dpj(~) sin(2n0)

÷ U2ijCf~i(~)~j('q) COS(2n0)

v (2) = vZ°4~i(~)chjOi) + vZS4~i(~)4~j(~) sin(Zn0)

+ v~Cchi(~)4~j(~) cos(Zn0)

w (2) = wZ°4h(~)4~j(~l) + w2S4ai(~)4~jOl) sin(Zn0)

+ wZC4~i(~)4~jO1) cos(2n0)

( i = 1 , 2 . . . . . p r ÷ l ; j : l , 2 . . . . . p x + l )

(9a-c)

This form of the solution is 'exact' in that it can be derived from the associated differential equations. The equilibrium equations governing the second-order field can now be gen- etically expressed in terms of the sets of degrees-of-freedom q~l) and ql 2) defining the first- and second-order fields, respectively:

{a,2) ,,.(2)'~ (2)__ ~ ~(1)~(1) -- AOij )qJ -- --t.irsttr tts

( i , j= 1,2 . . . . . N2; r , s = 1,2 . . . . . Ul) (10)

where Nz is the total number of degrees-of-freedom of the second-order displacement field. X is set equal to Xc~ in the solution of eqn (10).

Postbuckling response

In cases where a single mode governs the problem, it is

586


now possible to construct a potential energy function of the buckled structure in terms of the single scaling factor ~ in the form:

1 I I = ~(a - hb)~ 2 -'1- C~ 3 q -d~ 4 (11)

Note that c = 0 for the single mode problem considered here, i.e. the bifurcation is symmetric. The constants a and b are computed solely from the buckling mode, whereas the calculation of d involves both the buckling mode and the second-order field. In all the calculations the buckling mode which is normalized by prescribing the amplitude at the mid-section of the shell to be unity. ~, the scaling parameter, is identified with this amplitude. The equilibrium equation giving the postbuckling response may be written in the following non-dimensional form:

= 1 +X (2) with ~kcr = b; ~'(2)=4h2-da (12)

in which Xc~ is the critical value of the pressure and h is the shell thickness. The coefficient ~{2) is an index of the degree of imperfection sensitivity or possible postbuckling resistance of the structure. If X {2) is negative, the shell is imperfection-sensitive, i.e. the load-carrying capacity is reduced from the critical load in the presence of imperfections. If ~ is a measure of the imperfection in the form of the buckling mode and is defined in the same sense as ~, then an asymptotic estimate of the maximum load 12 is given by the solution to the cubic equation:

- S) 3 = 1 . 5 V / 3 ~ I ~ / h l S

with S = )kmax/~kcr; )k (2) <~ 0 (13)

Interactive buckling

Amplitude modulation. As a result of the interaction of the two fundamental modes of buckling, additional patterns of deformation are generated. These are given to the first-order accuracy by the mixed second-order field (~ 1,~ 2 field). It can be shown that this field consists principally of displacement fields in the form of local modes of buckling with wave numbers n - m and n + m, where n and m are the local and overall buckling wave numbers, respectively. When n >> m, the evaluation of this field with the standard pertur- bation procedure is riddled with singularities 4. This scenario is circumvented by the technique of amplitude modulation.

In the present treatment, an amplitude-modulated local mode is employed, i.e. the scalar parameter representing the magnitude of the local buckling displacements is allowed to vary in the circumferential and axial directions according to a 'slowly varying' function. Physically, the amplitude- modulating function accentuates the displacement amplitudes on the compression side of overall mode while at the same time decreasing the amplitude on the tension side. The amplitude-modulated local mode is equivalent to a linear combination of neighboring local modes with wave numbers . . . . n - m,n,n + m .... and has two effects: it

eliminates the singularity problem alluded to earlier and seriously diminishes the role of the mixed second-order field in the analysis. The mixed second-order field is now evaluated, by rendering the appropriate potential energy function stationary and imposing orthogonality with respect to the local buckling modes with n - m and n + m waves, respectively, which are included by proxy in the amplitude- modulated (first-order) local field.

Potential energy function for interactive buckling. The potential energy under the combined action of the two modes can now be developed. The displacements are taken in the form:

. (2) J ~.2 {u} = {u~1) t~1 + {4')l~(o)e~:(x)~i: + tu,11~

+ [u]Z)}fi(O)rbj(x)~o~ ~ + {u~Z)}f~(O)4~j(x)~O~kt (14)

where ~ 1 is the scaling factor for the overall buckling mode, ~ii represent the degrees-of-freedom associated with the amplitude modulation, and fi(O) and ~j(x) describe the variation of the amplitude-modulating function in the circumferential and radial directions, respectively. The superscript on u indicates the order of the field, subscripts 1 and 2 indicating, respectively, the local and overall quantities and a double subscript standing for the appropriate second- order field. ~:(x) is taken as linear over each bay andfi(O) is represented by a Fourier series consisting of at least three terms, i.e. axisymmetric, cos(m0) and sin(m0). This results in six degrees-of-freedom for each element (three on the left end of the element and three on the right end of the element). Over the junction and stiffener element q~ is taken as constant. Initial imperfections are included in the strains appropriately; employing Budiansky-Hutchinson notation 13

1 [e} = L 1 (U) q'- ~L2(u ) + Lll(U, ~) (15)

where u and a represent the displacement and initial imperfection vectors; the latter is taken in the form of buckling modes, local and/or overall. The potential energy function is evaluated by a procedure that exploits both the orthogonality of the fields involved and the 'slowly' varying nature of the overall fields and the modulating function. It may be expressed in the form:

1 2 + d 4 1I= - Xbl~l~l --Xbokl~ij~kl+ ~ ( a l - Xbl)~l 1~

-}- ~ (a i j k l - ~bijkl)~ij~k I --1-dijklpqrs~ij~kl~pq~rs

-}- eijkl~ij~kl~ l --I- gijkl~ij~kl~ 2 (16)

The first two linear terms on the right-hand side contain the contributions of initial imperfections in the overall and local modes respectively. The quadratic terms (the third and the fifth terms respectively) come from the buckling problem and the d' terms (the fourth and the sixth) encapsulate the postbuckling effects in individual modes. The e' and g' terms give the non-linear coupling between the modes. Of these, the cubic terms given by eUkl arise only when the amplitude modulation is considered. These are found to be

587


far more important than either the quartic interactive terms or the contributions of the mixed second-order field as discussed by Kasagi and Sridharan. The non-linear load- deflection path is generated by using an arc-length scheme 14.

A computer program INSTACC 15'16 has been written to perform all the analyses discussed here and has been placed in public domain.

Extension to the dynamic loading

In order to examine the effects of suddenly applied pressure, we may assume that the shell essentially responds by vibrating in the local and overall modes of buckling. Thus all the degrees-of-freedom in the static case are now treated as functions of time. Lagrange equations of motion are set up utilizing the potential energy function [eqn (16)] and setting up a simplified kinetic energy function that neglects the inertial contributions of the second-order fields. The response of the shell is now traced in the time domain for a given magnitude of suddenly applied pressure by using standard numerical techniques 17.

Non-linear analysis model

In order to examine the accuracy of the simplified methodology described here, full-blown non-linear analyses need to be performed incorporating appropriate initial imperfections. This involves large-scale computing often using commercial software and is beginning to be successfully undertaken TM. In the present study an attempt has been made to utilize ring elements with a harmonic description of the deformation in the circumferential direction in the spirit of studies reported here. The problem of non-linear coupling between the various harmonics is tackled by using a pseudo load technique in conjunction with a conjugate gradient scheme 19. A 2D version of the technique accounting for first-order shear deformation effects in the context of Sander's theory has been successfully developed 2°. Cur- rently, the package includes linear and non-linear bifurcation solvers as well as the full-fledged non-linear analysis for moderately thick composite cylinders. One of the drawbacks of the 2D analysis for ring-stiffened cylinders is that the stiffener-shell junctions are treated as point connections with the rigidity offered by the junction zone being lost--something that has serious consequences for modeling of the local buckling. Otherwise, the model can be a useful yardstick for investigating the accuracy of the asymptotic procedure presented here.

NUMERICAL EXAMPLES AND DISCUSSION

In this section we present results to (1) demonstrate the accuracy of the numerical results of INSTACC by appropriate comparisons with results available in the literature and those given by the 2D non-linear analyses and (2) to highlight aspects of shell behavior, such as the influence of lay-up sequence, imperfection sensitivity and collapse behavior.

Buckling analysis of shells under external pressure

Buckling analysis of angle-ply laminated shells has been reported by Simitses and associates 2'3 employing a Sander- type shell formulation in conjunction with two-dimensional, high-order shear deformation theory (HOSD) for which a cubic variation of u and v is considered in the transverse direction. The shell is assumed to be clamped at the ends and subjected to external lateral pressure only, i.e. the axial force P in Figure l(a) must be set to zero. The material properties (in psi, where applicable) are given below:

E l l = 3 0 X 106 E2z=E33=2 .7X 106

G12 = G13 = 0.65 X 106

G23 = 0.37 X 106 v12 = v13 = 0.21 l)23 = 0.45

In the present study the shell is modeled by two elements along the length. The prebuckling stresses are determined by the generalized plane strain condition, and the external lateral pressure is assumed to be centrally directed. The critical pressures for a set of R/h and L/R values and three different stacking sequences are given in Table 2. It is seen that our best results using p-levels, p , = 3 and Px = 5, generally yield slightly smaller values than those of Simitses et al. This is attributed to the way in which the shear deformations are considered. The present three-dimensional formulation provides a more rigorous description of shear deformations than the one based upon two-dimensional, high-order shear deformation theory. This observation becomes clear, in particular, from the results of the case of the short shell with LIR ---- 1. It is apparent that, despite the differences in the method of approach, the results obtained from the present study are in the range of good agreement.

Buckling and postbuckling analyses of composite shells

This example provides comparisons of buckling pressure and postbuckling response owing to the difference in lay-up patterns. Two families of relatively thick shells having R/h of 10 and 50, respectively, are considered. The material properties (in psi where applicable) are as follows:

Ell = 16 X 106 E22 =E33 = 1.48 X 106

Gl2 = G13 = 0.76 X 109 Pa

G23=0.51 x 106 v12=u13=0.33 t,23=0.45

The shells are discretized into two elements along the longitudinal direction and the analyses are carried out with the p-levels p, = 3 and Px ---- 5 for all cases. The shell is simply supported at its ends and treated as such for the determination of prebuckling stresses, and 'live' fluid pressure loading is considered.

Table 3 presents a comparison of buckling pressure for different stacking sequences. A [ 4 5 / - 45]QS laminate has the highest buckling resistance if the shell is short. This is an interesting phenomenon in that the coupling effects of

588


Table 2 Comparisons of buckling pressure for various laminated shells under external pressure, R I h = 15

[90/0/90]s k J E u )< 1016

L I R = 1 L I R = 2 L I R = 5

Present p-levels (Pr,Px) n = 4 n = 3 n = 2 (2,4) 5.186 2.407 1.326 (3,4) 4.891 2.366 1.320 (3,5) 4.881 2.362 1.312

Simitses et al. (HOSD) 4.897 2.364 1.300

[ - 4 5 / 4 5 / - 45]s ×or/Ell × 1016

L/R = 1 L/R = 2 L/R = 5

Present p-levels (Pr,Px) n = 8 n = 5 n = 3

(2,4) 4.930 2.366 0.862 (3,4) 4.515 2.268 0.851 (3,5) 4.508 2.260 0.844


[ 3 0 / 3 0 / - 60]~ X~/Ell × 1016

L/R = 1 L/R = 2 L/R = 5

Present p-levels (Pr,P~) n = 7 n = 6 n = 4 (2,4) 2.398 1.340 0.567 (3,4) 2.284 1.304 0.560 (3,5) 2.280 1.301 0.556


deformation generally reduce the stiffness of the laminate; however, it is surmised that the [ 4 5 / - 45]QS lay-up results in increased shear stiffness of the laminate and thus increases the buckling resistance to a small extent. Another observation is that a symmetric [90/0]s laminate has the highest buckling resistance to hydrostatic pressure in a wide range of the L/R ratio. This is not surprising in view of the following facts: (1) this laminate possesses the greatest bending stiffness in the circumferential direction in the family of layered shells considered herein and (2) the circumferential compression plays a major role in precipitating buckling. Table 4 provides the assessment of an orthotropic solution which treats an anisotropic laminate as an orthotropic one. Both [45/ - 45]OS and [45/ - 45]10 are antisymmetric, regular, angle-ply laminates, but the latter consists of five times as many laminae as the former. It is seen that (1) the buckling loads associated with a [ 4 5 / - 45]QS laminate are always smaller than those of a [45/-- 45] 10 laminate, and (2) the performance of the latter is very closely given by the corresponding orthotropic solutions. This is because of the abating of the adverse coupling effects as the number of plies is increased. So, in these cases the orthotropic solutions may be justified. For example, at L/R = 0.5, the [45/ - 45] 10 shell gives only 0.5% smaller buckling pressure than the orthotropic one as against a 12% smaller value associated with the [45/--45]Q s shell. Nevertheless, the difference between anisotropic and orthotropic solutions decreases as the length of the shell increases.

The postbuckling response of laminated composite shells, represented by the value of X (2), is illustrated in Figure 3(a) Figure 3(b). It is found that the postbuckling characteristics vary with the laminate lay-up pattern; the differences are

clearly observed in the lower range of L/R. The shells become less imperfection-sensitive as the length increases and buckle in neutral equilibrium like Euler columns. In Figure 3(a) a [30/ - 60]s laminate exhibits a slightly different postbuckling response from the others and turns out to be the most imperfection-sensitive except in the lower range of L/R ratio. Reading this together with Table 3(a) in which the stacking sequence [ 3 0 / - 60]~ shows the lowest buckling resistance over the entire range of L/R ratio, this lay-up pattern has little practical interest.

Interactive buckling

Two types of cylinder, having stacking sequences [90/0]s (family #1) and [ 4 5 / - 45]QS (family #2) for the shell skin are considered. In both cases the stiffeners are assumed to be made up of [90/0] s laminates. The end boundary conditions are assumed to be simply supported. The material properties are the same as in Section 3.2.

We define the volume ratio p such as:

p = V s t i f f / V t o t a l

in which Vtota I is the total volume of material used for the shell including the stiffeners, whereas Vs~fe is that of stiffeners alone. In the present study, Vtotal and P are kept constant as the number of stiffeners (Ns) varies.

In order to capture the most severe effects of interaction, some care must be bestowed in the selection of the local mode. The local mode, if highly localized near the ends, would not interact with the overall mode to any appreciable extent. Such 'edge modes' are discarded in this study in favor of a local mode that is pervasive, exhibiting significant deformation at or near the center. Table 5 presents the geometric parameters [refer to Figure l(c) for notation] as well as the buckling pressures (hi and X2 are the critical pressures for overall and local buckling) and the corresponding postbuckling coefficients (X(2)). All the significant values of pressure are rendered dimensionless by Eu. It is seen that the shell having 10 stiffeners (Ns = 10) is the case where two critical pressures are coincident, triggering both the local and the overall modes simultaneously. In this case, an edge (local) mode corresponding to a lower critical

5 pressure X2/Eu = 1.63 × 10- has been passed over in favor of a higher pervasive mode with ×2/Ell = 1.89 × 10 -5. All the computations are performed with p-levels, Pr = 2 and p x = 5 .

Imperfection-sensitivity surface. The following discussion deals with the case of simultaneous buckling which high- lights the main features of interactive buckling. Figure 4(a) Figure 4(b) gives the imperfection-sensitivity surface for coincident buckling (Ns = 10), showing the maximum load-carrying capacity, km~x, for a given combination of imperfection parameters. ~1 and E2 are dimensionless amplitudes of overall and local imperfection modes, respectively, and are obtained by dividing the corresponding imperfections by the radius and the thickness of the shell, respectively. If any one of the two imperfection parameters is set to zero, i.e. along the paths ~1 = 0 or E2 = 0, the

589


Table 3 Comparisons of buckling pressure for various laminates

L/R k~/Ell × 10 4 (n) 0 . 0 - 0 . 1 .

[90/0]s [ 3 0 / - 60]~ [ 4 5 / - 45]s [ 4 5 / - 45]QS -0.2.

(a) Case R/h = 10 -0.3. 1 8.37(3) 8.16(5) 10.2(5) 11.3(5) -0.4- 2 5.72(3) 3.87(4) 4.84(4) 5.07(3) -0.5" 4 3.40(2) 1.89(3) 2.40(3) 2.48(3) k(2) -0.6 • 6 2.49(2) 1.06(2) 1.28(2) 1.21(2) -0.7- 8 2.24(2) 0.764(2) 0.996(2) 0.949(2) -0.8 10 2.15(2) 0.660(2) 0.888(2) 0.861(2) -0.9

-1.0- L/R )kcr/Ell × 105 (n) -1.1

-1.2 [90/0]s [30/-- 60]~ [ 4 5 / - 45]~ [45/-- 45]Q s

(b) Case R / h = 5 0 0.5 3.58(7) 3.30(9) 4.39(10) 5.04(10) 1 2.37(5) 1.47(7) 1.87(7) 1.91(7) 2 1.44(4) 0.669(5) 0.831(5) 0.795(5) 4 0.817(3) 0.302(4) 0.402(4) 0.380(4) 6 0.566(3) 0.184(3) 0.237(3) 0.219(3) 8 0.508(3) 0.143(3) 0.199(3) 0.185(3)

10 0.350(2) 0.129(3) 0.142(2) 0.134(2)

Table 4 Comparisons between anisotropic and orthotropic solutions for R/h = 50

LIR hcr/Ell × 105 (n)

[ 4 5 / - 45]QS [ 4 5 / - 45] l0 Orthotropic solution

0.5 5.04(10) 5.68(10) 5.71(10) 1 1.91(7) 2.12(7) 2.13(7) 2 0.795(5) 0.880(5) 0.883(5) 4 0.380(4) 0.423(4) 0.425(4) 6 0.219(3) 0.242(3) 0.242(3) 8 0.185(3) 0.206(3) 0.206(3) 10 0.134(2) 0.142(2) 0.142(2)

primary solution path loses the stability by bifurcation at a point where a secondary coupled equilibrium path inter- sects, triggering deformation in the mode quiescent so far. These figures exhibit a severe reduction of load-carrying capacity, kmax, in the range of small imperfections. In contrast, the rate of load reduction becomes smaller as imperfection magnitudes are increased.

Significance of amplitude modulation. An examination of the deformation of the shell circumferentially and longi- tudinally indicates that the local buckling amplitudes are accentuated in regions which buckle inwards (thus causing additional circumferential compression) due to the overall buckling and by the same token tend to be very small in regions where the shell buckles outwards under the influence of overall buckling. Evidently, the compression induced due to inwards bending associated with overall buckling is responsible for accentuation of the buckles. This represents a significant feature of the behavior of the stiffened shell undergoing interactive buckling and must not be ignored in any analysis.

Extensive numerical studies 4 have shown that a relatively simple but sufficiently accurate interaction analysis is possible considering the set of cubic terms ~ ij~ 0f i alone to depict the interaction in the potential energy function and neglecting all the remaining interactive terms. This term can be computed without involving any of the second-order

(a) C a s e R / h - - 1 0

" / R/h = 10 / r .-

o" / ... - - [90/01s

/ ..-" _ - [45/-45]s

.- . . . . . [45/-451Q s

- . - [30/-601s

5

L/R

(b) Case R/h = 50 0 . 0 -[ - - ~ . . . . . - .

/ -°"1 - 0 3 f / _..'" R / h = 5 0

" "'" ~ [ 9 0 / 0 ] h(2) -0 .4~ / :'" S

[ . . . " .. _ _ [45/-45]S ~ 0 ~ 5

] . . . " ..... [45/-45]Q s

10

0.5 5 10

L/R

Figure 3 Postbuckling response of composite shells under hydrostatic pressure. (a) Case Rlh = 10; (b) case R/h = 50

fields, i.e. solely in terms of the buckling modes and the amplitude-modulating function. Interestingly, this term would have vanished but for the amplitude modulation.

Interlaminar stresses. The interlaminar stresses play a key role in precipitating delamination-type failure in composite shells. It is found that the magnitude of transverse shearing stresses is far higher than the normal stress, arr, as the gradient of w in the radial direction is found to be extremely small; attention is therefore focussed on the former in the sequel. Figure 5 plots the stresses r~0 and r~x versus a load parameter k for the shell family #1 with 10 stiffeners. The inteflaminar stresses are evaluated at a location where the respective components take the maximum values: r,0 is at the center of the shell while ~'rx is at the junction to a stiffener nearest from the center. The figure shows that the rate of growth of the stress increases drastically in the vicinity of the load limit point. These stresses appear to be significant enough to be a factor in design.

Spectra of imperfection sensitivity. In this section the imperfection sensitivity of shells is discussed with the ratio of the critical pressures, X2/kh varied over a rather wide range. For a given total volume of the shell and geometric proportions of the stiffeners, the number of stiffeners, Ns, controls the ratio of critical pressures. Figure 6(a) Figure 6(b) presents the variation of the maximum load with Ns for three sets of imperfections of increasing magnitude. The curves marked by X 1 and )k 2 indicate the critical loads of

590


1.0 1.0

0.8

0.6

0.4

0.2

1.0

0.8 0.8

0.6

0.4

0.6

0.4

0.2 0.2

"7 4"]0 ~ 10 - lo , \ o

Figure 4 Imperfection sensitivity of (a) [90/0]s ring-stiffened shell with XI/X2 ~ 1 and (b) [ 4 5 / - 45]QS ring-stiffened shell with ~%1/~%2 ~ 1

respective buckling modes, see Table 5. The dashed lines give the maximum loads attainable without the interaction as given by the asymptotic analysis [eqn (13)].

The figures show that the shells are very imperfection- sensitive in the range where X x and ~k 2 a r e close each other. In this region the interaction curves lie much below the dashed lines. Imperfection sensitivity is rather more severe in the region in which the local buckling load is smaller than the overall one. This is due to the fact that the shells possess more strongly negative posthuckling coefficient h (2) with respect to the local mode of buckling than with respect to overall buckling. Thus as the imperfections increase, the maximum load attainable shifts towards the region where hi < h2. An optimum point exists on the side in which Ns > 10, moving away from the simultaneity point towards the region where the overall mode is the governing mode--so much so, that a design based on classical 'naive' criterion is not necessarily optimal. There is clear advantage in making the local critical pressure higher than the overall critical pressure by 50% 0~2/XI ~ 1.5). Such shells are clearly less imperfection- sensitive than those with coincident critical values and thus retain much of their capacities in the presence of imperfections. Furthermore, the severe interlaminar stresses that usually accompany local buckling are reduced.

The spectra reveal that the effects of interaction are most significant in the range of critical stress ratios (Xz/X1) of 1/2 to 2. In this range the reduction of the load-carrying capacity can be as high as 50% for imperfections of such magnitude as may be unavoidable in practice.

Dynamic modal interaction

The stiffened shell (shell #2) is further investigated to study the effects of dynamic loading and mode interaction on the buckling pressure of the shell. Figure 7 illustrates these reductions when considering only overall imperfections in the interactive analysis with minute local imperfections to circumvent a dynamic bifurcation. The top two curves show the reductions as found from a single-mode analysis while the bottom two curves show the reduction due to the interactive analysis. In the range of imperfections investigated, the curves show that the load reduction increases monotonically with imperfection magnitudes.

1.0

g

0.8

0.6

0.4

0 . 2

0 0

- - TrO

. . . . . "¢ rx

~1=0.01; ~2=0. l

/ 0.5 1.0

h x 105

1.5

Figure 5 Maximum interlaminar stresses v e r s u s pressure for a shell under interactive buckling

591


3.0

2 .5

2.0

1.5

1.0

0.5

( a )

::::::::::::::::::::::::: ...............

• • Overall mode

[] [] Local mode

. . . . . . Single mode analysis

Interaction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1: .ml=0.001; ~2=0.01

2: ~1=0.005; ~2=0.05

3: ~1=0.01; ~2=0.1

I I 9 10 11

N s

3.0

2.5

2.0

1.5

1.0

0.5

0 12

7.0

6.5

6.0

5.5

5.0

4.5

'~ 4.0

3.5

Figure 6

3,0

2.5

2.0

1.5

( b ) • • Overall mode

[] [ ] Local mode

. . . . . . . Single mode analysis

Interaction k 2

7.0

6.5

6.0

5.5

" f 5.0 4.5

. . . . . . . . . . . . . . . . 4 . 0 _ 3.5

. . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . 2

I I I 8 9 10 11

N s

Imperfection-sensitivity spectrum for (a) the [90/0]s shell family and (b) the [45/- 45]QS]s shell family

3 "1- 3,0

I: ~1=0.001; ~2=0.01

2: -~1=0.005; ~2=0.05

3: ~1=0.01; ~2=0.1 t 2.5

2.0

1.5 12

Comparison with 2D model

In order to check the accuracy of the results produced by INSTACC on the response of stiffened shells, buckling and

full-blown single-mode non-linear analyses were performed by using 2D models. In particular, this is expected to be of value in assessing the accuracy of the asymptotic procedure which performs a pivotal role in the interactive buckling analysis.

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Table 5 Buckling pressure and postbuckling response of the stiffened shells, R = 50 in (dominant mode of buckling underlined)

N~ d/ts S~IR Overall mode Local mode

~.cr/Ell × 10 5 (n) X (2) hcr/Ell × 10 5 (n) X (2)

(a) [90/0], shells, L /R= 2.62, R/h = 100, tJh = 2.0, p --- 28% 8 3.414 0.274 2.49(3) + 0.010 1.48(10) - 0.106 9 3.022 0.244 2.15(3) 1.69(10) 10 2.711 0.220 1.90(3) - 0.007 1.89(10) - 0.047 11 2.458 0.200 1.71(3) 2.10(10) 12 2.248 0.183 1.57(3) - 0.002 2.30(10) + 0.050

N~ d/ts SplR Overall mode Local mode

X~r/Ell X 10 5 (n) X (23 X J E l l X 10 5 (n) )l (2)

(b) [ 4 5 / - 45]QS shells, L/R = 4.64, R/h = 62.5, tslh = 1.5, p = 27% 8 6.708 0.494 5.77(2) 9 5.900 0.442 5.04(2) 10 5.267 0.400 4.58(2) 11 4.756 0.365 4.27(2) 12 4.337 0.335 4.03(2)

- 0.106 3.22(10) - 0.353 3.88(11)

- 0.054 4.59(11) - 0.518 5.39(11)

- 0.023 6.23(12) - 0.708

The following two shells are selected for the study:

• Case ( i ) - -a composite shell belonging to family #1 (RIh = 100, Ns = 10; see Table 5) having near-coincident critical pressures subjected to 'live' pressure as before.

• Case ( i i )--a shell of isotropic material analyzed by Moradi and Parsons 21 with the following properties: E = 1 0 4 0 0 k s i , v = 0.3, L = 2.38in, h = 0.023in, t~ = 0.017 in, R = 4.00 in, Ns = 6, the ratio of volume of stiffener material to that of shell = 0.24 (based on centerline dimensions). The loading is assumed to be 'dead' pressure.

The buckling pressures are computed by using both linear stability analysis and non-linear bifurcation analysis with the prebuckling non-linearities duly considered in the latter. The results are presented in Table 6 and Table 7.

First consider the overall buckling predictions. It is seen that for case (i) (Table 6), the predictions of 2D linear stability and INSTACC agree very well. The non-linear bifurcation analysis produces a bifurcation pressure that is higher than the linear prediction by about 4%. This is in marked contrast to the behavior of thinner shell, i.e. case (ii) (R/h = 174). In this case the critical pressures based on linear and non-linear bifurcation analyses differ significantly, with the latter giving a value which is about 12% smaller. It may be concluded that, provided the shells are sufficiently thick (RIh < 100) and buckle in a long-wave mode, linear stability analysis may provide an acceptable conservative estimate for the critical pressure.

Next consider the local buckling pressures of case (i). The pressures reported are the lowest and these correspond to

Table 6 Comparison of critical pressures: case (i)

Type of model Overall buckling Local buckling

(hl/E]t) X 10 "5 m (X2/En) × 105 n

INSTACC (3D) 1.90 linear stability 2D linear stability 1.89 2D non-linear 1.98 bifurcation

3 1.63 10

3 1.55 10 3 1.27, 1.58 a 10

°With local thickening in the first bay

'edge modes'--localized near the edges. There is a noticeable difference (5%) in the local buckling pressures predicted by the 2D models on the one hand and INSTACC on the other. Because of the fact that the 2D models work with the center-line dimensions, the width-to-thickness ratio of the bay is increased by 20% in this case and this is a factor in the reduction of the predicted local pressures. Thus 2D predictions of local buckling pressures may not be sufficiently accurate.

Again, for the case of local buckling the prebuckling non- linearities play a significant role. This is because they are concentrated in the end zones and hasten the local buckling with a mode localized near the supports. This effect can, however, be eliminated by a local thickening of the shell over the end bays. This is illustrated in the Table 6, where it is seen that, by a increase of shell thickness of 20% (0.6 in instead of 0.5 in) over the end bay alone, the non-linear bifurcation pressure is significantly enhanced and attains a value higher than that predicted by the linear stability analysis.

The composite shell is analyzed for the non-linear response by using the 2D model with imperfection only in the overall mode (.~.l = 0.01). Only three harmonics are included, namely 0, 3 and 6, to account for axisymmetric, buckling and postbuckling responses. This precludes the intervention of the local mode. The shell response is typified

Table 7 Comparison of critical pressures: case (ii)

Type of model Buckling pressure (psi) Overall mode (m = 7)

3D linear stability INSTACC 302 ABAQUS (20-node brick elements) 307

2D linear stability Present 302 BOSOR4 302 ABAQUS (eight-noded quadrilaterals) 303

2D non-linear bifurcation Present 265 BOSOR4 265 ABAQUS (eight-noded quadrilaterals) 265

593


Figure 7

0.9.

0.8.

0.7-

Ii 0.6-

0.5- 0.4- 0.3

0

J • ~ 6 c l n l c r ~ i w I • dymmaic iateraaive

• static single mode

,* dyna~c ~ l c mode

i l l I I

1 2 3 4 5 I I I I

6 7 8 9 10

Overall Imperfection, El • 103

Loss of load-carrying capacity due to modal interaction and dynamics

by the variation of the amplitude of the principal harmonic (m = 3) at the center with k. This is compared with X versus ~1 given by the asymptotic procedure (3D model, INSTACC) in Figure 8. The two results agree closely in the earlier stages of loading but deviate noticeably in the advanced stages. The maximum value of the load as given by the full-fledged non-linear analysis is about 9% higher. This is not surprising as the asymptotic procedure ceases to be accurate for larger imperfection magnitudes; typically, imperfections must be a small fraction of the thickness.

Thus it may be inferred that for thick shells with moderate levels of imperfections, the asymptotic procedure gives a conservative and sufficiently accurate prediction of the maximum load.

It is instructive to examine the shell response if the interaction with local buckling is considered. To illustrate this, we select a pervasive local mode previously identified (see Section 3.3) to be the key mode in the interaction. The shell response with local buckling acting alone with an initial imperfection of ~2 = 0.1 is included in Figure 8. Tiffs

270.000 ////4' 240.000 /

/ /

! 210.000 /

i 180.000 !

I

"~ lso.ooo i

E , I~ 120.OO0 . i - - 7 - / [ ;

90.0°° i ; / / / 6o. ; /

30.000 /

00.000 0.00 0.15

. . . . . . . Overall Budding Response ONSTACC) . . . . Local Buckling Rcspon~ (INSTACC)

Interacfiw Buckling ~ S T A C C ) - - - Nonlinear Analysis In Ovm~ Mod~

0.30 0.45 0.60 0.75 0.90 1.05 1.20 1.35 1.50 1£5

~I , ~z , in.

Figure 8 Comparison of load-displacement relationships as given by the asymptotic approach and fully non-linear analysis

594


response is given in terms of the local buckling amplitude measured at the center of the shell (~2) and was obtained by using the asymptotic procedure. Finally, the interactive c,j,~u: buckling response under combined imperfection is also d:

indicated in the same figure in terms of ~ 1. It is seen that h: under one mode active at a time, the shell exhibits far L: greater load-carrying capacity for the same level of LbLH, L2: m, n: imperfections; and the mode interaction abruptly destabi- lizes the shell at about 55% of the critical load of the Ns: shell. P:

q0: The asymptotic procedure captures well the load- R, RbRo:

deflection behavior (as evidenced by its good agreement r, x: with fully non-linear analysis) in the range of loading where sp:

ts: failure occurs by interaction. This underlines the accuracy u, v, w: of the asymptotic analysis for the purposes of incorporation into the interactive analysis. Greeks

CONCLUSION ANDFUTUREWORK

A powerful algorithm for postbuckling analysis and modal interaction based on 3D elasticity has been developed for composite ring-stiffened cylindrical shells. The shells can lose up to 50% of their buckling capacity owing to modal interaction in the presence of relatively small imperfections. The phenomenon of modal interaction is in essence governed by amplitude modulation and any procedure that does not account for it explicitly or implicitly cannot capture the destabilization that occurs due to modal interaction.

The linear stability analysis employed to extract the buckling loads and modes is seen to provide a conservative estimate of buckling pressures for relatively thick shells as far as overall buckling is concerned. In the case of local buckling, prebuckling non-linearities can precipitate a localized form of local buckling at a significantly smaller pressure than given by the linear stability analysis. If local buckling is a competing mode of failure, local thickening over the end bays would help prevent premature collapse of the shell.

The asymptotic procedure incorporated in the interactive analysis is seen to be accurate at least up to the range where collapse occurs as a result of mode interaction. In the absence of modal interaction, the asymptotic procedure itself can give estimates of acceptable accuracy of the maximum load.

For ring-stiffened shells of moderate thickness, two- dimensional analyses that do not account for the rigidity offered by the shell-stiffener junctions underestimate significantly the local buckling pressures. A thorough exploration of the relative accuracies of the various models is urgently needed.

The present work has focused exclusively on linear elastic material response. For thick composite shells with R/h < 20 (say), material non-linearity induced typically by the shear softening of the matrix can be an important factor in the prediction of buckling pressures and postbuckling response. This is a subject of current study at Washington University.

NOTATION

Ex, EO, Er: "~r O, ~/rx, ~/Ox:

0: k:

Xl, k2: ),max:

~1, ~2:

~2: II: (...)~% (...)<', (...)% ('"h, ('"h:

('"h2:

elastic constants stiffener depth shell thickness shell length linear, bilinear and quadratic operators, respectively number of circumferential waves for the overall and local modes of buckling, respectively number of stiffeners axial force external lateral pressure mean, inner and outer radii of shell, respectively radial and longitudinal coordinates clear stiffener spacing stiffener thickness displacement components in the x-, 0- and r-directions, respectively

normal strain components (engineering) shear strain components natural coordinates in the radial and longitudinal directions circumferential coordinate load parameter (pressure) critical pressure posthuckling coefficient overall and local buckling pressures, respectively maximum load-carrying capacity scaling factor for buckling deformation scaling factors (amplitudes at the center) of overall and local modes, respectively imperfection parameters for overall and local modes, respectively degrees-of-freedom depicting the amplitude-modulating function

= ~dR = ~2/h

potential energy function prebuckling quantities first- and second-order field quantities, respectively quantifies associated with the overall and local mode, respectively quantities of the mixed second-order field

ACKNOWLEDGEMENTS

The present work is supported by the Office of Naval Research Grant No. N00014-91-J1637. The authors are grateful to Dr Yapa D.S. Rajapakse for his encouragement.

REFERENCES

1. Kardomateas, G.A. Buckling of orthotropic cylindrical shells under external pressure. J. Appl. Mech., ASME 1993, 195-202.

2. Simitses, G.J. and Anastasiadis, J.S., Shear deformable theories for cylindrical laminates--equilibrium and buckling with applications. AIAA J., 1992, 30(3), 826-834.

3. Simitses, G.J., Tabiei, A. and Anastasiadis, J.S., Buckling of moderately thick laminated cylindrical shells under lateral pressure. Compos. Eng., 1993, 3(5), 409-417.

4. Kasagi, A. and Sridharan, S., Modal interaction in composite cylinders under hydrostatic pressure. Int. J. Solids Struct., 1995, 32(10), 1349-1369.

5. Koiter, W.T. and Pignataro, M. A general theory of the interaction between local and overall buckling in stiffened panels, Report WTHD-83, Delft University of Technology, Delft, Holland, 1976.

6. Jones, R.M. 'Mechanics of Composite Materials', Hemisphere Pub- lishing, New York, 1975.

7. Ambartsumyan, S.A. 'Theory of Anisotropic Plates', Technomic Publishing, Westport, CO, 1970.

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8. Szab6, B.A. and Babugka, I. 'Finite Element Analysis', John Wiley and Sons, New York, 1991.

9. Kasagi, A. and Sridharan, S., Buckling and posthuckling analyses of thick composite cylindrical shells under hydrostatic pressure using axisymmetric solid elements. Compos. Eng., 1993, 3(5), 467-487.

10. Brush, D.O. and Almroth, B.O. 'Buckling of Bars, Hates and Shells', McGraw-Hill, New York, 1975.

11. Koiter, W.T. General equations of elastic stability for thin shells. In 'Proceedings of Symposium on the Theory of Shells to Honor Lloyd Hamilton Donnell', University of Houston, TX, 1967, pp. 187-228.

12. Koiter, W.T. 'On the stability of equilibrium' (in Dutch), Thesis, Polytechnic Institute, Delft, Holland, 1945 (English translation: NASA, Tech. Trans., F-10,833, 1967).

13. Budiansky, B, and Hutchinson, J.W. Dynamic buckling of imperfection-sensitive structures. In 'Proceedings of l lth Inter- national Congress of Applied Mechanics', Spfinger-Verlag, Berlin, 1964.

14. Crisfield, M.A., A fast incremental/iterative solution procedure that handles snap-through. Computers and Structures, 1981, 13, 55-62,

15. Kasagi, A. 'Interactive buckling of ring stiffened composite cylindrical shells', D.Sc. Thesis, Department of Civil Engineering, Washington University, St. Louis, MO, 1994.

16. Kasagi, A., Sridharan, S. and Schokker, A.J. 'INSTACC User Manual', Department of Civil Engineering, Washington University, St. Louis, MO, 1994.

17. Schokker, A., Kasagi, A. and Sridharan, S. Dynamic interactive buckling of ring stiffened composite shells. A1AA J., 1995, 33, 1954-1962.

18. Bonnani, D.L. Stability analysis of thick-section composite cylinders under hydrostatic pressure including three-dimensional effects and nonlinear material response. In 'Compression Response of Composite Structures', ASTM STP 1185, (Eds. S. Groves and A.L. Highsmith), American Society for Testing Materials, Philadelphia, PA, 1994, pp. 7-33.

19. Ravichandran, R.V., Sridharan, S. and Gould, P.L., A local-global model for the nonlinear analysis of locally defective shells of revolution. Int. J. Num. Meth. Eng., 1994, 37, 3057-3074.

20. Sridharan, S. and Alberts, J. Some aspects of numerical modeling of buckling offing-stiffened composite shells. AIAA J., 1997, 35, 187- 195.

21. Moradi, B. and Parsons, I.D. A comparison of modelling techniques for the buckling of stiffened shells. In 'Proceedings of ASCE Engineering Mechanics Specialty Conference', Columbus, OH, 1991, pp. 856-860

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On the buckling and collapse of moderately thick composite cylinders under hydrostatic pressure

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