Postbuckling of 3D braided composite cylindrical shells under combined external pressure and axial compression in thermal environments

Postbuckling of 3D braided composite cylindrical shells under combined external pressure and axial compression in thermal environments

ARTICLE IN PRESS International Journal of Mechanical Sciences 50 (2008) 719–731 www.elsevier.com/locate/ijmecsci Postbuckling of 3D braided composit...

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ARTICLE IN PRESS

International Journal of Mechanical Sciences 50 (2008) 719–731 www.elsevier.com/locate/ijmecsci

Postbuckling of 3D braided composite cylindrical shells under combined external pressure and axial compression in thermal environments Zhi-Min Lia, Hui-Shen Shena,b, a

School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China

b

Received 26 December 2006; received in revised form 3 December 2007; accepted 6 December 2007 Available online 14 December 2007

Abstract A postbuckling analysis is presented for a three-dimensional (3D) braided composite cylindrical shell of finite length subjected to combined loading of external pressure and axial compression in thermal environments. Based on a micro–macro-mechanical model, a 3D braided composite may be a cell system and the geometry of each cell is highly dependent on its position in the cross-section of the cylindrical shell. The material properties of epoxy are expressed as a linear function of temperature. The governing equations are based on a higher order shear deformation shell theory with a von Ka´rma´n–Donnell-type kinematic nonlinearity and includes thermal effects. A singular perturbation technique is employed to determine interactive buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, braided composite cylindrical shells with different values of shell geometric parameter and of fiber volume fraction under combined loading conditions. The results show that the shell has lower buckling loads and postbuckling paths when the temperature-dependent properties are taken into account. The effects of temperature rise, fiber volume fraction, shell geometric parameter, load-proportional parameter, as well as initial geometric imperfections are studied. r 2008 Elsevier Ltd. All rights reserved. Keywords: 3D braided composites; Cylindrical shell; Thermal effect; Postbuckling; Combined loading

1. Introduction In recent years, fiber-reinforced composite shell structures have been widely used in the aerospace, marine, automobile and other engineering industries. In some of these applications, the cylindrical shell structures may be subjected to external pressure combined with axial compression. The capability to predict the response of composite shells when subjected to combined loads is of prime interest to structural analysis. A new class of composite materials known as textile composites has drawn considerable attention. Textile composites are manufactured by fabrication methods derived from the textile industry. Unlike laminated Corresponding author at: School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China. E-mail address: [email protected] (H.-S. Shen).

0020-7403/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2007.12.001

composites, in which cracking and debonding may occur at a high temperature due to the material property mismatch at the interface of two discrete materials [1,2], the textile composites are able to eliminate the delamination due to the inter-lacing of the tows in the throughthickness direction. Numerous investigations for determining their physical, mechanical and thermal properties are available in the literature; see, for example, [3–6]. Many buckling and failure studies have been conducted for thin and moderately thick composite cylindrical shells subjected to pure axial compression [7–12], or uniform external pressure [13–16]. However, investigations involving the application of the shear deformation shell theory to the postbuckling analysis of textile composite cylindrical shells under combined loadings are limited in number. Kaddour et al. [17] reported experimental results for the failure behavior of 7551 filament-wound E-glass/epoxy tubes subjected to combined external pressure and axial compression with various ratios of circumferential to axial stress.

ARTICLE IN PRESS Z.-M. Li, H.-S. Shen / International Journal of Mechanical Sciences 50 (2008) 719–731

720

P0

γ F B

R

D E

A

B C

F A E

ϕi

Cell A

Cell B

Cell C

Cell D

L

q

X′ h X

Z Cell E

Y

Cell F

Z′ O′

Y′

P0 Fig. 1. A braided composite cylindrical shell under combined lateral pressure and axial compression.

Dong and Mistry [18], and Mistry et al. [19] presented an experimental and numerical investigation for 7551 filament-wound glass/epoxy pipes subjected to combined external pressure and axial compression. Consequently, Mistry [20] examined the wind angle effects in filamentwound glass/epoxy pipes and concluded that the failure response is dependent upon the combination of the loads and the wind angle of the fiber reinforcements. In the aforementioned studies, filament-wound and woven shells may be merger the two-dimensional (2D) preforms from industrial textile processes, e.g. knitting, weaving or braiding. These preforms offer good responses to plane stress states. However, to obtain thicker composite parts or more complex shapes, it becomes necessary to stack or join different 2D layers, and coexistence in a laminate of stiff and strong plies with a relatively weak matrix interface is really harmful. There may be damage, like delamination or debonding, when the matrix is principally loaded under fatigue, shear or after some impact. Consequently, there is a need to develop a three-dimensional (3D) model for braided composites. Zeng and Wu [21] presented a postbuckling analysis for a stiffened braided composite cylindrical thin shell subjected to combined loading of external pressure and axial compression. In their analysis, a 3D braided three-cell model was used and a boundary layer theory for the shell buckling suggested by Shen and Chen [22,23] was adopted. In their studies, the shells were considered as being relatively thin and, therefore, the transverse shear deformation was not accounted for. In fact, the loaded braiding cells may play a great role in moderately thick shells. However, studies on postbuckling of shear deformable braided composite cylindrical shells subjected to combined loading of external pressure and axial compression in thermal environments and with temperature-dependent material properties have not been found in the literature. In the present study, we focus on the 3D braided composite cylindrical shells subjected to combined loading

of external pressure and axial compression. A 3D braided composite model is established, from which four interior cells and two surface cells are obtained (see Fig. 1). We assume that the cross-section of multifilament braiding yarns is elliptical and all yarns in the braided preform have identical constituent material, size and flexibility. The temperature field considered is assumed to be a uniform distribution over the shell surface and through the shell thickness. The material properties of epoxy are expressed as a linear function of temperature. The governing equations are based on Reddy’s higher order shear deformation shell theory with a von Ka´rma´n–Donnelltype kinematic nonlinearity and including thermal effects. A singular perturbation technique is employed to determine interactive buckling loads and postbuckling equilibrium paths. Both the nonlinear prebuckling deformations and initial geometric imperfections of the shell are taken into account. The solution methodology is similar to that described in [24], but the solutions are not repeated here for convenience. The numerical illustrations show the full nonlinear postbuckling response of braided composite cylindrical shells under the combined action of external pressure and axial compression in thermal environments. 2. Theoretical development A traditional 2D triaxial braided composite laminate of a certain desired thickness is formed by overbraiding layers, each with a specified braiding angle that can be serially superimposed in order to form a multi-layer braid. Unfortunately, the problem with these multi-layer braids lies in the interlaminar weakness or in other words in its sensitivity to delamination. To overcome this problem, the 3D braiding technique, in which the braiding yarns interlock through a volume of material, is developed. To successfully predict the mechanical properties of 3D braided composites, realistic geometrical unit cells, which

ARTICLE IN PRESS Z.-M. Li, H.-S. Shen / International Journal of Mechanical Sciences 50 (2008) 719–731

take into account the effect of yarn configuration, are required. Based on the movement of carriers, we analyzed the yarn traces systematically in 3D tubular braided preforms using the control volume and control surface method similarly reported by Wang and Wang [5]. A new micro–macromechanical model of unit cells was established. The macrocell of the model is further decomposed into simpler elements, called unit cells here, whose stiffness is calculated from the geometry of the fiber tow and the stiffness of the constituents and yarn cross-section changes into an elliptic shape due to interaction between two yarns. According to the load-sharing relations between the unit cells, the macroscopic stiffness is then calculated. In this model, a 3D braided composite may be a cell system and the geometry of each cell is highly dependent on its position in the crosssection of the cylindrical shell. We assume that the yarn (fiber tow) is transversely isotropic and the matrix is isotropic, from which the stiffness matrix can be expressed as   ½C ¼ V f ½T C f ½TT þ V m ½C m , (1) where Vf and Vm are the yarn and the matrix volume fractions and are related by Vf þ Vm ¼ 1

(2)

and [Cf] and [Cm] are stiffness matrixes for the yarn and the matrix, respectively, and can be expressed as 3 2 0 0 0 cf 11 cf 12 cf 13 6 cf 22 cf 23 0 0 0 7 7 6 7 6 6 cf 33 0 0 0 7   6 7 (3a) Cf ¼ 6 7, cf 44 0 0 7 6 7 6 6 Sym cf 55 0 7 5 4 cf 66 2 6 6 6 6 6 ½C m  ¼ 6 6 6 6 4

cm11

cm12 cm22

cm13 cm23

0 0

0 0

cm33

0

0

cm44

0

Sym

cm55

3 0 7 0 7 7 0 7 7 7 0 7 7 0 7 5

(3b)

ð1  vf 12 vf 21 ÞE f 1 ð1 þ vf 23 Þvf 21 E f 1 ; cf 12 ¼ , D D ð1  vf 12 vf 21 ÞE f 2 ; cf 13 ¼ cf 12 , cf 33 ¼ cf 22 ¼ D ðvf 23 þ vf 12 vf 21 ÞE f 2 ; cf 55 ¼ G f 13 ¼ cf 66 ¼ G f 12 , cf 23 ¼ cf 44 ¼ D D ¼ 1  2vf 12 vf 21 ð1 þ vf 23 Þ  v2f 23 ,

cm44 ¼ cm55 ¼ cm66

Em ¼ Gm ¼ , 2ð1 þ nm Þ

l 21

6 2 6 l2 6 6 l2 6 ½T ¼ 6 3 6 l2l3 6 6l l 4 31 l1l2

m21

n21

2m1 n1

2n1 l 1

m22

n22

2m2 n2

2n2 l 2

m23

n23

2m3 n3

2n3 l 3

m2 m3 m3 m1

n2 n3 n3 n1

m2 n3 þ m3 n2 m3 n1 þ m1 n3

n2 l 3 þ n3 l 2 n3 l 1 þ n1 l 3

m1 m2

n1 n2

m1 n2 þ m2 n1

n1 l 2 þ n2 l 1

2l 1 m1

3

7 7 7 7 2l 3 m3 7 7, l 2 m3 þ l 3 m2 7 7 l 3 m1 þ l 1 m3 7 5 l 1 m2 þ l 2 m1 2l 2 m2

(4) where lk, mk, nk (k ¼ 1, 2, 3) are the direction cosines of a yarn with braiding angle g and inclination angles b, where b is the angle between the projection of yarn axis on the Y0 O0 Z0 plane and the Y0 -axis, and the braiding angle g is the angle between the yarn axis and the X0 -axis, as shown in Fig. 1, and can be defined as l 1 ¼ cos g; l 2 ¼ sin g cos b; l 3 ¼ sin g sin b; m1 ¼ 0; m2 ¼ sin b; m3 ¼  cos b; n1 ¼  sin b;

n2 ¼ cos g;

(5a)

n3 ¼ cos g sin b

and g ¼ ðg1 þ g2 Þ=2, gj ¼ tan1 ð4Lj =hÞ

ðj ¼ 1; 2Þ,

Lj ¼ 2riþj1 sinðp=2NÞ sinðjiþj1 þ p=2NÞ þ d=2 tan jiþj1 ðj ¼ 1; 2Þ,

riþ1 ¼ ri þ d=ð2 sin ji Þ;

r1 ¼ Din =2 þ d; r2m1 ¼ Dout =2  d

ði ¼ 1; 2;    ; 2M  2Þ,

cf 11 ¼

cm12 ¼ cm13 ¼ cm23 ¼

2

b ¼ p=2  ji ,   ji ¼ a cos ½d cosðp=NÞ=½4ri sinðp=2NÞ þ p=2N,

in which

Em ; 1  v2m

where Ef1, Ef2, Gf12, nf12 and nf23 are the Young’s modulus, shear modulus and Poisson’s ratio, respectively, of the fiber, and Em, Gm and nm are corresponding properties for the matrix. In Eq. (1) [T] is the transformation matrix from the local coordinate system to the global coordinate system and can be expressed as

d ¼ h tan gi cos ji =2,

cm66

cm11 ¼ cm22 ¼ cm33 ¼

721

vm E m , 1  v2m

(3c)

(5b)

where ji is the solid cross-sectional orient angle of the yarn, M, N are referred to [M  N], respectively, being the numbers of radical columns and circumference rows of the braiding carriers, Din and Dout are the inner and outer diameters of the cylindrical shell and d is the short axis of cross-sectional of the yarn, Lj is the horizontal projective length of braiding yarn and the value of pitch length h can be measured from the preform h iexterior. Therefore, the transformed elastic constants Qij can be

written as h i 1 Z c Z t=2 Qij ¼ ½C dz dy ct 0 t=2

(6)

ARTICLE IN PRESS Z.-M. Li, H.-S. Shen / International Journal of Mechanical Sciences 50 (2008) 719–731

722





















  and Qij

from which the ‘‘lamina’’ stiffnesses can be calculated by

where

ðAij ; Bij ; Dij ; E ij ; F ij ; H ij Þ X Z c=2 Z zl   1 Mþ2 ¼ Qij ð1; z; z2 ; z3 ; z4 ; z6 Þ dz dy 2 2c l¼4 0 tl Z Z X c=2 tlþ1   1 Mþ2 þ Qij ð1; z; z2 ; z3 ; z4 ; z6 Þ dz dy 1 2c l¼4 0 zl Z Z M  c=2 zl  1X þ Qij ð1; z; z2 ; z3 ; z4 ; z6 Þ dz dy 3 2c l¼2 0 tl Z Z M  c tlþ1  1X þ Qij ð1; z; z2 ; z3 ; z4 ; z6 Þ dz dy 4 2c l¼2 0 zl M Z c Z zl   X 1 þ Qij ð1; z; z2 ; z3 ; z4 ; z6 Þ dz dy 1 2c l¼2 c=2 tl M Z c Z tlþ1   X 1 þ Qij ð1; z; z2 ; z3 ; z4 ; z6 Þ dz dy 2 2c l¼2 c=2 zl Z c Z zl   Mþ2 X 1 þ Qij ð1; z; z2 ; z3 ; z4 ; z6 Þ dz dy 4 2c l¼4 c=2 tl Z c Z tlþ1   Mþ2 X 1 þ Qij ð1; z; z2 ; z3 ; z4 ; z6 Þ dz dy 3 2c l¼4 c=2 zl Z t=2þts      1 Qij þ Qij ð1; z; z2 ; z3 ; z4 ; z6 Þ dz dy þ 5 6 2 t=2 Z 1 t=2      þ Qij þ Qij ð1; z; z2 ; z3 ; z4 ; z6 Þ dz dy ði; j ¼ 1; 2; 6Þ, 5 6 2 t=2ts

are referred to stiffness matrix of four interior cells (see Fig. 1 cell A–D) and two surface cells (see Fig. 1 cell E–F) in different positions. In Eqs. (6) and (7) (with l=2, 3,?, M+2)

(7a) ðAij ; Dij ; F ij Þ ¼

X 1 Mþ2 2c l¼4

Z 0

c=2

Z

zl tl



Qij

 2

ð1; z2 ; z4 Þ dz dy

X Z c=2 Z tlþ1   1 Mþ2 Qij ð1; z2 ; z4 Þ dz dy 1 2c l¼4 0 zl Z Z M   c=2 z l 1X þ Qij ð1; z2 ; z4 Þ dz dy 3 2c l¼2 0 tl Z Z M   c t lþ1 1X þ Qij ð1; z2 ; z4 Þ dz dy 4 2c l¼2 0 zl Z Z M   c z l 1X þ Qij ð1; z2 ; z4 Þ dz dy 1 2c l¼2 c=2 tl Z Z M   c t lþ1 1X þ Qij ð1; z2 ; z4 Þ dz dy 2 2c l¼2 c=2 zl X Z c Z zl   1 Mþ2 þ Qij ð1; z2 ; z4 Þ dz dy 4 2c l¼4 c=2 tl X Z c Z tlþ1   1 Mþ2 þ Qij ð1; z2 ; z4 Þ dz dy 3 2c l¼4 c=2 zl Z t=2þts      1 Qij þ Qij þ ð1; z2 ; z4 Þ dz dy 5 6 2 t=2 Z    1 t=2   þ Qij þ Qij ð1; z2 ; z4 Þ dz dy ði; j ¼ 4; 5Þ, 5 6 2 t=2ts þ

(7b)

Qij

1

;

Qij

2

;

Qij

3

;

Qij

4

c ¼ h tan gl , zl ¼ y= tan jl þ ð2l  M  7Þd=2 sin jl

;

Qij

5

6

ð0pypcÞ,

zl ¼  y= tan jl þ ð2l  M  3Þd=2 sin jl

ð0pypcÞ,

tl ¼ ð2l  M  7Þd=2 sin jl , tl ¼ ð2l  M  3Þd=2 sin jl , ts ¼ ðt  ðM  1Þd= sin jl Þ=2.

(8)

Consider a circular cylindrical shell made of 3D braided composites. The length, mean radius and total thickness of the shell are L, R and t. The shell is assumed to be relatively thick, geometrically imperfect and is subjected to two loads combined out of a uniform lateral pressure q and axial compressive load P0 (see Fig. 1). The shell is referred to a coordinate system (X, Y, Z), in which X and Y are in the axial and circumferential directions of the shell and Z is in the direction of the inward normal to the middle surface. The corresponding displacements are designated by U; V and W . Cx and Cy are the rotations of normal to the middle surface with respect to the Y-and X-axes, respectively. The origin of the coordinate system is located at the end of the shell on the middle plane. Denoting the  initial deflection by W ðX ; Y Þ, let W ðX ; Y Þ be the additional deflection and F ðX ; Y Þ be the stress function for the stress resultants defined by N x ¼ F ;yy ; N y ¼ F ;xx and N xy ¼ F ;xy ; where a comma denotes partial differentiation with respect to the corresponding coordinates. Reddy and Liu [25] developed a simple higher order shear deformation shell theory, in which the transverse shear strains are assumed to be parabolically distributed across the shell thickness and which contains the same dependent unknowns as in the first-order shear deformation theory, and no shear correction factors are required. This shell theory was widely used in many shell analyses for moderately long as well as for short shells [26,27]. Based on Reddy’s higher order shear deformation theory, and using von Ka´rma´n–Donnell-type kinematic relations, the governing differential equations for 3D braided composite cylindrical shells subjected to external pressure combined with axial compression are derived and can be expressed in terms of a stress function F , two rotations Cx and Cy and transverse displacement W , along with initial geometric imperfection  W . They are       T L~ 11 W  L~ 12 Cx  L~ 13 Cy þ L~ 14 F  L~ 15 N    ¯ T  1 F¯ ;xx ¼ L~ W þ W  ; F þ q, (9)  L~ 16 M R

ARTICLE IN PRESS Z.-M. Li, H.-S. Shen / International Journal of Mechanical Sciences 50 (2008) 719–731

  1     T L~ 21 F þ L~ 22 Cx þ L~ 23 Cy  L~ 24 W  L~ 25 N þ W ;xx R 1 ~  ¼  L W þ 2W ; W , (10) 2

    L~ 31 W þ L~ 32 Cx þ L~ 33 Cy þ L~ 34 F    T T  L~ 35 N  L~ 36 S ¼ 0,

(11)

    L~ 41 W þ L~ 42 Cx þ L~ 43 Cy þ L~ 44 F    T T  L~ 45 N  L~ 46 S ¼ 0,

(12)

(13a) and 2 T3 2 T3 2 T3 Sx Mx Px 6 T7 6 T7 7 4 6 6S 7 6M 7 6 PT 7 6 y 7 ¼ 6 y 7  2 6 y 7, 4 5 4 T 5 3t 4 T 5 T M xy Pxy Sxy

Q12 Q22 Q26

32 2 l1 6 2 6 m Q26 7 54 1 2l 1 m1 Q66 Q16

(13b)

l 22 m22 2l 2 m2

3 " # 7 a11 7 5 a22 (14)

in which a11 and a22 are the thermal expansion coefficients measured in the fiber and transverse directions, respectively, and defined by [28] a11 ¼

V f E f 1 af 1 þ V m E m am , V f Ef 1 þ V mEm

a22 ¼ V m ð1 þ vm Þam þ V f ð1 þ vf 12 Þaf 2  v12 a11 ,

Z

(15b)

(16a)

2pR

N x dY þ 2pRtsx þ pR2 qa ¼ 0, 0

2pR

2 q2 F 4  qCx  q F  þ A þ B  E 12 21 21 3t2 qX qX 2 qY 2 0



2 2 4  qCy 4   q W  q W þ B22  2 E 22 þ E 22  2 E 21 3t 3t qY qX 2 qY 2 #

2    W 1 qW qW qW T T    þ   A12 N x þ A22 N y dY ¼ 0. R 2 qY qY qY

Z

A22

(17b) Because of Eq. (17), the in-plane boundary condition V ¼ 0 (at X ¼ 0, L) is not needed in Eq. (16). The average end-shortening relationship is Z 2pR Z L Dx 1 qU dX dY ¼  2pRL 0 L 0 qX Z 2pR Z L 1 q2 F q2 F ¼  A11 þ A12 2 2pRL 0 qY qX 2 0



4  qCx 4  qCy   þ B11  2 E 11 þ B12  2 E 12 3t 3t qX qY



2 4 q2 W q2 W 1 qW þ E 12  2 E 11  2 3t 2 qX qX qY 2

   qW qW T T    A11 N x þ A12 N y dX dY .  (18) qX qX It is assumed that Em is a function of temperature, then all braiding stiffnesses Aij, Bij, Dij, etc., are also functions of temperature. Furthermore, h i h ini Eqs. h (17b) i h and i h(18)ithe reduced h i stiffness matrices Aij ; Bij ; Dij ; E ij ; F ij and H ij ,

(i, j=1, 2, 6) are functions of temperature, determined through relationships [24] A ¼ A1 ;

(15a)

where af1, af2 and am are thermal expansion coefficients of the fiber and matrix respectively. The two end edges of the shell are assumed to be clamped, so that the boundary conditions are X ¼ 0, L: W ¼ Cx ¼ Cy ¼ 0,

where a ¼ 0 and 1 for the lateral and hydrostatic pressure loading cases, respectively, and sx is the average axial compressive stress, M x is the bending moment and Px is the higher order moment, as defined in Ref. [25]. Also we have the closed (or periodicity) condition Z 2pR qV dY (17a) qY 0 or

~ Þ are where linear operators L~ ij ð Þ and nonlinear operator Lð defined in Ref. [24]. Note that the geometric nonlinearity in ~ Þ in Eqs. (9) the von Ka´rma´n sense is given in terms of Lð and (10). The thermal forces, moments and higher order moments caused by temperature change DT are defined by 2 T 3 T T 2 3 N x M x Px Ax Z M 6 T 7 t 6 N M T PT 7 X k 6 A 7 6 y 4 y 5 ð1; Z; Z 3 ÞDT dZ y y 7¼ 4 T 5 t k¼1 k1 A T T xy N xy M xy Pxy k

where 2 3 2 Ax Q11 6A 7 6Q 4 y 5 ¼ 4 12 Axy Q16

723

(16b)

B ¼ A1 B;

D ¼ D  BA1 B,

E ¼ A1 E; F ¼ F  EA1 B;

H ¼ H  EA1 E.

(19)

It is worth noting that the governing differential equations (9)–(18) for a 3D braided composite shell are identical in form to those of unsymmetric cross-ply laminated shells. Hence, we may employ a singular perturbation technique to determine the interactive buckling loads and postbuckling equilibrium paths. The solutions are obtained in the same form as previously reported in Ref. [24]. Such solutions are easy to program and the numerical results can readily be obtained. The major difference herein is that the shell stiffnesses are determined based on a micro-mechanical model. Note that in the case of combined loading, two kinds of loading

ARTICLE IN PRESS Z.-M. Li, H.-S. Shen / International Journal of Mechanical Sciences 50 (2008) 719–731

724

conditions, i.e. case 1 and case 2, should be considered. For case 1, a high value of external pressure is combined with relatively low axial load, and for case 2, a high value of axial compression is combined with relatively a low external pressure.

axial compression. The results are listed in Table 1 and compared with the experimental results of Kaddour et al. [17]. The computing data adopted here are Ef1 ¼ 72.4 GPa, Ef2 ¼ 71.3 GPa, Gf12 ¼ 28.6 GPa, nf12 ¼ 0.25 GPa, Em ¼ 4.0 GPa and nm ¼ 0.35 and fiber volume fraction Vf ¼ 0.68. As a second example, we examine the buckling pressure qcr (in MPa) for filament-wound shells subjected to combined lateral pressure and axial compression. The results are listed in Table 2 and compared with the experimental results of Mistry et al. [19], using their material properties, i.e. Ef ¼ 72 GPa, nf ¼ 0.25, Em ¼ 3.8 GPa, nm ¼ 0.39 and Vf ¼ 0.55. Note that in these two tables sy/sx ¼ 1:0 means pure lateral pressure and sy/sx ¼ 0:1 means pure axial compression. As we all know, the Donnell-type approximations are accurate when the circumferential wave

3. Numerical results and discussions Numerical results are presented in this section for perfect and imperfect, 3D braided composite cylindrical shells with clamped boundary conditions subjected to combined loading of external pressure and axial compression. We first examine the interactive buckling loads qcr (in MPa) and Pcr (in kN) for 7551 filament-wound E-glass/ epoxy tubes subjected to combined lateral pressure and

Table 1 Comparison of critical external pressure (qcr) (MPa) and axial compression Pcr (kN) for 7551 GRP tubes subjected to combined external pressure and axial compression t

Test no.

5 1 16 36 11 43 65 38 17 61 48 44 62 64 63 18 7 31 8

R/t

9.43 9.57 4.78 4.94 4.65 4.76 4.66 4.67 4.78 7.13 9.595 9.9 5.87 5.2 4.24 4.78 4.73 9.51 4.64

5.80 5.725 5.835 5.662 5.984 5.857 5.972 5.96 5.835 4.076 5.711 5.551 4.844 5.404 6.514 5.835 5.89 5.76 6.00

L/R

6.76 6.75 6.63 6.61 6.65 6.64 6.65 6.65 6.63 6.36 6.75 6.73 6.51 6.58 6.70 6.63 6.64 6.76 6.65

Z

265.33 261.11 256.72 247.70 264.52 257.90 263.90 263.29 256.72 165.15 260.36 251.65 205.05 234.23 292.25 256.72 259.67 262.91 265.14

sy/sx

0:1 0:1 1:2 1:1 1:1 2.5:2 1.5:1 3:2 3:2 1.7:1 2.4:1 2.4:1 2.5:1 2.5:1 2.7:1 3:1 4:1 5:1 6:1

(qcr, Pcr) Test [17]

Present

(0, 495) (0, 494) (14.14, 122.3) (40.4, 117) (37.7, 109) (59.12, 103) (77.12, 37.74) (74.99, 71.1) (66.9, 63.4) (135.7, 67.2) (82.69, 159.8) (89.45, 172.4) (84.57, 130.4) (71.29, 109.2) (74.0, 63.47) (79.9, 76.8) (55.68, 81.8) (42.6, 293) (39.39, 79.2)

(0, 569.77) (0, 567.68) (15.79, 131.06) (42.69, 122.37) (44.32, 112.25) (61.45, 108.42) (80.05, 44.39) (79.86, 78.52) (71.56, 69.81) (136.17, 75.19) (86.48, 169.47) (89.98, 180.24) (85.43, 137.67) (72.22, 115.26) (80.34, 64.02) (81.13, 94.51) (57.66, 89.20) (47.38, 298.08) (43.28, 83.91)

Table 2 Comparison of critical external pressure (qcr) (MPa) for filament-wound shells subjected to combined lateral pressure and axial compression Sample number

L/R

sy/sx ¼ 1:0

sy/sx ¼ 2:1

sy/sx ¼ 1:1

sy/sx ¼ 1:2

Testa [19]

Present

Test [19]

Present

Test [19]

Present

Test [19]

Present

1

4.88 9.76 19.5

2.5 1.2 0.7

2.52 (1, 2)b 1.22 (1, 2) 0.78 (1, 2)

2.4 1.1 0.7

2.47 (1, 3) 1.46 (1, 2) 0.82 (1, 2)

2.0 1.0 0.7

2.27 (1, 3) 1.06 (1, 2) 0.75 (1, 2)

2.0 1.0 0.7

2.28 (1, 3) 1.03 (1, 2) 0.78 (1, 2)

2

9.52 19.0 38.1

5.4 5.0 4.4

5.48 (1, 2) 5.11 (1, 2) 4.44 (1, 2)

6.2 5.2 4.9

6.27 (1, 2) 5.48 (1, 2) 4.97 (1, 2)

5.5 4.9 5.1

5.73 (1, 2) 5.28 (1, 2) 5.15 (1, 2)

5.2 4.1 3.1

5.43 (2, 2) 4.34 (2, 2) 3.29 (2, 2)

3

4.84 9.68 19.4

4.5 1.8 1.3

4.50 (1, 3) 2.03 (1, 2) 1.45 (1,2)

4.4 1.9 1.3

4.43 (1, 2) 1.96 (1, 2) 1.35 (1, 2)

4.1 2.0 1.3

4.18 (1, 2) 2.15 (1, 2) 1.30 (1, 2)

3.6 1.5 1.2

3.86 (2, 3) 1.77 (1, 2) 1.44 (1, 2)

a

Values read from graphs of [19]. The numbers in brackets indicate the buckling mode (m, n).

b

ARTICLE IN PRESS Z.-M. Li, H.-S. Shen / International Journal of Mechanical Sciences 50 (2008) 719–731

number is greater than 2. From Tables 1 and 2, it can be seen that the agreement of the present results with those of Kaddour et al. [17] and Mistry et al. [19] is in general not particularly good. Note that in several cases the observed discrepancy is quite higher than 5%. This is due to the fact that the shell may have initial geometric imperfections in the experiments. In Table 2, (m, n) indicates the buckling mode, which determines the number of half-waves in the X-direction and of full waves in the Y-direction. In addition, the load-end-shortening curves for a (745) woven composite glass/epoxy cylindrical shell subjected to axial compression alone are compared in Fig. 2 with the test results (DF06) and finite element method (FEM) results of Spagnoli et al. [12]. In Ref. [12] the numerical

Present (W*/t = 0.0)

2

(W*/t = 0.065) Spagnoli et al. [12] 1: EXP, Pcr = 232 kN 2: FEM, Pcr = 386 kN

P (kN)

300

simulations were carried out by using finite element program ABAQUS. The computing data adopted are L ¼ 600 mm, R ¼ 301.86 mm, t ¼ 3.72 mm, Ef ¼ 69 GPa, nf ¼ 0.23, Em ¼ 4.0 GPa, nm ¼ 0.34 and Vf ¼ 0.308. It can be seen that the present solution for the perfect shell is much better than the FEM results. Moreover, the pressure–axial strain curves for a 7551 filament-wound E-glass/epoxy tube subjected to combined lateral pressure and axial compression (sy/sx ¼ 6:1) are compared in Fig. 3 with the test results (test no. 8) of Kaddour et al. [17]. The computing data adopted here are the same as in Table 1. The results calculated show that when an initial geometric  imperfection is present, e.g. W =t ¼ 0:065 for Fig. 2 and  W =t ¼ 0:3 for Fig. 3, the present results are in reasonable agreement with the experimental results. Table 3 Comparisons of buckling loads (scr, qcr) (in MPa) for perfect braided cylindrical shells with different values of shell geometric parameter and fiber volume fraction subjected to combined lateral pressure and axial compression in thermal environments (R/t ¼ 40)

500

400

Vf

L/R

Z

DT ¼ 0 K

DT ¼ 100 K

DT ¼ 200 K

0.4

1.581

100

(190.719, 0) (185.917, 0.166) (126.703, 0.564) (0, 1.010)

(182.980, 0) (178.363, 0.159) (122.095, 0.547) (0, 0.979)

(175.210, 0) (170.779, 0.152) (119.118, 0.530) (0, 0.949)

2.236

200

(190.564, 0) (185.099, 0.165) (93.035, 0.414) (0, 0.669)

(182.857, 0) (177.605, 0.158) (90.202, 0.401) (0, 0.649)

(175.122, 0) (170.083, 0.151) (87.364, 0.389) (0, 0.596)

3.535

500

(190.407, 0) (184.531, 0.164) (65.294, 0.290) (0, 0.418)

(182.156, 0) (177.094, 0.158) (63.238, 0.282) (0, 0.405)

(174.452, 0) (169.630, 0.149) (61.180, 0.272) (0, 0.392)

1.581

100

(232.135, 0) (226.276, 0.201) (154.717, 0.688) (0, 1.218)

(225.629, 0) (219.935, 0.196) (151.566, 0.674) (0, 1.207)

(219.107, 0) (213.575, 0.190) (148.409, 0.661) (0, 1.182)

2.236

200

(231.954, 0) (225.289, 0.201) (113.533, 0.505) (0, 0.817)

(225.482, 0) (219.004, 0.195) (111.176, 0.495) (0, 0.800)

(218.994, 0) (212.702, 0.189) (108.816, 0.485) (0, 0.783)

3.535

500

(231.251, 0) (224.607, 0.199) (79.564, 0.354) (0, 0.509)

(224.803, 0) (218.373, 0.194) (77.853, 0.347) (0, 0.498)

(218.350, 0) (212.125, 0.189) (76.142, 0.339) (0, 0.488)

1.581

100

(273.538, 0) (266.623, 0.237) (182.727, 0.813) (0, 1.456)

(268.269, 0) (261.495, 0.233) (180.210, 0.802) (0, 1.435)

(262.990, 0) (256.358, 0.228) (177.691, 0.790) (0, 1.415)

2.236

200

(273.332, 0) (265.468, 0.236) (134.028, 0.596) (0, 0.964)

(268.096, 0) (260.392, 0.232) (132.145, 0.588) (0, 0.949)

(262.852, 0) (255.306, 0.227) (130.259, 0.580) (0, 0.937)

3.535

500

(272.656, 0) (264.673, 0.236) (93.832, 0.417) (0, 0.601)

(267.449, 0) (259.641, 0.231) (92.465, 0.412) (0, 0.592)

(262.233, 0) (254.602, 0.226) (91.097, 0.405) (0, 0.583)

1 200

100

0 0

1

2 3 4 End-shortening (mm)

5

Fig. 2. Comparisons of postbuckling load-end shortening curves for a woven composite glass/epoxy cylindrical shell under pure axial compression.

0.5

60 σy/σx = 6 : 1

q (MPa)

40

0.6 Present W*/t = 0.0

20

W*/t = 0.3 Kaddour et al. [17] EXP: Test No.8

0 0

5000

10000

725

15000

Axial strain (10-6) Fig. 3. Comparisons of postbuckling pressure–axial strain curves for a filament-wound glass/epoxy tube under lateral pressure combined with axial compression.

ARTICLE IN PRESS Z.-M. Li, H.-S. Shen / International Journal of Mechanical Sciences 50 (2008) 719–731

726

A parametric study has been carried out and the typical results are listed in Table 3 and shown in  Figs. 4–13. In all these figures, W =t denotes the dimensionless maximum initial geometric imperfection of the shell. For these examples R/t=40 and the total thickness of the shell is t=4 mm, pitch length h=2.8 mm and M  N=9  400. The carbon fiber tows are selected as the braiding material and the material properties of carbon fiber and epoxy adopted, as given in Refs. [28–30], are: Ef1=201.0 GPa, Ef2=13.5 GPa, Gf12=9.5 GPa, nf12= 0.22 GPa, nf23=0.377, af11=0.7  106 K1, af22=8.0  106 k1, Em=(4.3–0.003DT) GPa and nm=0.34 and am=40.0  106 K1. The buckling loads (scr, qcr) (in MPa) for perfect braided composite cylindrical shells under four sets of combined

loading conditions, i.e. lateral pressure alone (b1=0), combined loading case (1) (b1=10), combined loading case (2) (b2=0.02) and axial compression alone (b2=0), and under three sets of environmental conditions, i.e. 1: DT=0 K, 2: DT=100 K, 3:DT=200 K, are calculated and compared in Table 3, in which the shell geometric parameter Z ¼ 100; 200 and 500, and three values of the fiber volume fraction Vf ¼ 0.4, 0.5 and 0.6 are considered. It can be seen that, for a braided composite cylindrical shell, the buckling loads are reduced with increases in temperature and with decreases in fiber volume fraction. Fig. 4 shows the effects of temperature rise and fiber volume fraction on the interaction buckling curves of a braided composite cylindrical shell under combined loading cases, in which Rq ¼ q/qcr and Rp ¼ sx/scr, where qcr

1.2

1.2 Z = 500, R/t = 40, Vf = 0.5

1:Vf = 0.4 2:Vf = 0.5 3:Vf = 0.6 3

0.8 2

Rp

0.8

Rp

1 2 3

1

0.4

0.4

0.0

0.0 0.4

0.0

1.2

0.8

0.4

0.0

0.8

1.2

Rq

Rq

Fig. 4. Effects of temperature rise and fiber volume fraction on the interaction buckling curves of a braided composite cylindrical shell subjected to combined axial compression and hydrostatic pressure.

2500

2500

I: b2 = 0.0 II: b2 = 0.02

2000

I: b2 = 0.0 II: b2 = 0.02

I

T-ID

2000

T-D II

1500 Px (kN)

Px (kN)

1500

1000

T-ID

1000

T-D I II

W*/t = 0.0

500

500

W*/t = 0.0

W*/t = 0.1

W*/t = 0.1

W*/t = 0.3

W*/t = 0.3

0

0 -2

0

2

4 Δx (mm)

6

8

10

0.0

0.5

1.0

1.5

2.0

W/t

Fig. 5. Effect of temperature dependency on the postbuckling behavior of a braided composite cylindrical shell subjected to axial compression combined with hydrostatic pressure. (a) Load–shortening and (b) load–deflection.

ARTICLE IN PRESS Z.-M. Li, H.-S. Shen / International Journal of Mechanical Sciences 50 (2008) 719–731

2500

2500

2000

2000 I

I: b2 = 0.0 II: b2 = 0.02

1500

727

I: b2 = 0.0 II: b2 = 0.02

1500 Px (kN)

Px (kN)

II

1 3 2

1000

1000

I&1,2,3 II&1,2,3

500

500

W*/t = 0.0

W*/t = 0.0 W*/t = 0.1

W*/t = 0.1

0

0 -2

0

2

4 Δx (mm)

6

8

10

0.0

0.5

1.0

1.5

2.0

W/t

Fig. 6. Effect of temperature rise on the postbuckling behavior of a braided composite cylindrical shell subjected to axial compression combined with hydrostatic pressure. (a) Load–shortening and (b) load–deflection.

2500

2500 R/t = 40, Z = 500 Vf = 0.5, (m, n) = (6, 5) 1: Vf = 0.4 2: Vf = 0.5 3: Vf = 0.6

2000

I&3

2000 II&3 I&2

I: b2 = 0.0 II: b2 = 0.02

1500

R/t = 40, Z = 500 Vf = 0.5, (m, n) = (6, 5) 1: Vf = 0.4 2: Vf = 0.5 3: Vf = 0.6

1500 Px (kN)

II&2 I&1

Px (kN)

I: b2 = 0.0 II: b2 = 0.02

II&1

1000

1000 I&3 II&3 II&2 I&2 I&1 II&1

500

500

W*/t = 0.0

W*/t = 0.0 W*/t = 0.1

W*/t = 0.1

0

0 -2

0

2

4 Δx (mm)

6

8

10

0.0

0.5

1.0

1.5

2.0

W/t

Fig. 7. Effect of fiber volume fraction on the postbuckling behavior of a braided composite cylindrical shell subjected to axial compression combined with hydrostatic pressure. (a) Load–shortening and (b) load–deflection.

and scr are the critical buckling loads under lateral pressure alone or axial compression alone. Note that in Fig. 4(a) these critical values are for the shell with Vf ¼ 0.5 at DT ¼ 0 K, whereas in Fig. 4(b) they are for the shell with Vf ¼ 0.6 at DT ¼ 100 K. It is seen that fiber volume fraction has a significant effect, whereas the temperature variation only has a small effect on the shape of the interaction buckling curves. Fig. 5 presents the postbuckling load–shortening and load–deflection curves for a braided composite cylindrical

shell with Z ¼ 500 and Vf ¼ 0.5 under DT ¼ 100 K and under two cases of thermoelastic material properties, i.e. T-ID and T-D, subjected to combined loading case (2) with the load-proportional parameter b2 ¼ 0.0 (referred to as I) and 0.02 (referred to as II). Here, TD indicates that Em is temperature dependent, whereas TID indicates that Em is temperature independent, i.e. Em ¼ 4.3 GPa. It can be seen that the postbuckling equilibrium paths become lower when the temperature-dependent properties are taken into account. The postbuckling load–shortening and

ARTICLE IN PRESS Z.-M. Li, H.-S. Shen / International Journal of Mechanical Sciences 50 (2008) 719–731

728

2500

2500

R/t = 40, Vf = 0.5, ΔT = 100 K 1:Z = 100, (m, n) = (3, 5) 2:Z = 200, (m, n) = (4, 5) 3:Z = 500, (m, n) = (6, 5)

2000

R/t = 40, Vf = 0.5, ΔT = 100 K 1:Z = 100, (m, n) = (3, 5) 2:Z = 200, (m, n) = (4, 5) 3:Z = 500, (m, n) = (6, 5)

2000 II&2

II&1 I&1

I&1

I&3

II&1

I&2

1500

1500

II&2

Px (kN)

Px (kN)

II&3

I&2 I&3

1000

1000

500

500

W*/t = 0.0

I: b2 = 0.0 II: b2 = 0.02

II&3

W*/t = 0.0

I: b2 = 0.0 II: b2 = 0.02

W*/t = 0.1

W*/t = 0.1

0

0 -2

0

2

4 Δx (mm)

6

10

8

0.0

0.5

1.0

1.5

2.0

W/t

Fig. 8. Effect of shell geometric parameter on the postbuckling behavior of braided composite cylindrical shells subjected to axial compression combined with hydrostatic pressure. (a) Load–shortening and (b) load–deflection.

1.0

1.0

R/t = 40, Z = 500 Vf = 0.5, (m, n) = (6, 5)

0.9

0.9

0.8

0.8

0.7

0.7 I

1: Z = 100 2: Z = 200 3: Z = 500

0.6

I&1 II&1 I&2 II&2 I&3 II&3

II

I: b2 = 0.0 II: b2 = 0.02

I&1 II&1 I&2 II&2 I&3 II&3

I

0.6

II I: b2 = 0.0 II: b2 = 0.02

0.5

0.5 0.0

0.1

0.2

0.3

W*/t

0.0

0.1

0.2

0.3

W*/t

Fig. 9. Comparison of imperfection sensitivities of braided composite cylindrical shells subjected to axial compression combined with hydrostatic pressure.

load–deflection curves for imperfect braided composite  cylindrical shells (W =t ¼ 0:1 and 0:3) have been plotted,  along with the perfect shell (W =t ¼ 0) results, in Fig. 5. It can be seen that for a small imperfection the buckling loads drop down and the elastic limit load for an imperfect shell can be achieved and imperfection sensitivity can be predicted, whereas for a larger imperfection the postbuckling equilibrium path becomes stable. Fig. 6 shows the effect of temperature rise DT ( ¼ 0, 100 and 200 K) on the postbuckling load–shortening and load– deflection curves for the same braided composite cylindrical

shell under combined loading case (2). It is found that an initial extension occurs as the temperature increases and the buckling loads are reduced with increases in temperature. Fig. 7 shows the effect of fiber volume fraction Vf( ¼ 0.4, 0.5 and 0.6) on the postbuckling load–shortening and load–deflection curves for a braided composite cylindrical shell with Z ¼ 500 under DT ¼ 100 K and under a combined loading case (2). It can be seen that the buckling loads are reduced with decreasing fiber volume fraction, and the postbuckling equilibrium path becomes lower when Vf is decreased.

ARTICLE IN PRESS Z.-M. Li, H.-S. Shen / International Journal of Mechanical Sciences 50 (2008) 719–731

1.2

1.2

I: b1 = 0.0 II: b1 = 10.0

1.0

I: b1 = 0.0 II: b1 = 10.0

1.0

0.8

0.8

0.6

T-ID T-D

I

q (MPa)

q (MPa)

729

T-ID T-D

0.4

0.6

T-ID T-D T-ID T-D

I

0.4

II

II

0.2

W*/t = 0.0

0.2

W*/t = 0.0

W*/t = 0.1

W*/t = 0.1

0.0

0.0 -2

-4

0

2

4

0

6

2

Δx (mm)

4

6

8

W/t

Fig. 10. Effect of temperature dependency on the postbuckling behavior of a braided composite cylindrical shell subjected to hydrostatic pressure combined with axial compression. (a) Load–shortening and (b) load–deflection.

1.2

1.2

1.0

1.0

I: b1 = 0.0 II: b1 = 10.0

0.6 I&2

I&3

I: b1 = 0.0 II: b1 = 10.0

0.8 q (MPa)

q (MPa)

0.8

I&1 II&3

II&2

II&1

0.4

0.6

I

1

23 12 3

0.4 II

0.2

W*/t = 0.0

0.2

W*/t = 0.0

W*/t = 0.1

W*/t = 0.1

0.0

0.0 -6

-4

-2

0

2 4 Δx (mm)

6

8

10

0

2

4

6

8

W/t

Fig. 11. Effect of temperature rise on the postbuckling behavior of a braided composite cylindrical shell subjected to hydrostatic pressure combined with axial compression. (a) Load–shortening and (b) load–deflection.

Fig. 8 shows the effect of shell geometric parameter Z ( ¼ 100, 200, 500) on the postbuckling load–shortening and load–deflection curves of braided composite cylindrical shells with Vf ¼ 0.5 under DT ¼ 100 K and under a combined loading case (2). The results show that the slope of the postbuckling load–shortening curve for the shell with Z ¼ 100 is larger than others, and the shell has considerable postbuckling strength. Fig. 9 gives imperfection sensitivity l for the imperfect braided composite cylindrical shells with Vf ¼ 0.5 and

different values of shell geometric parameter Z ( ¼ 100, 200 and 500) under a combined loading case (2) with the load-proportional parameter b2 ¼ 0.02 and under thermal environmental conditions DT ¼ 0, 100 and 200 K. Here, l is the maximum value of sx for the imperfect shell, made dimensionless by dividing by the critical value of sx for the perfect shell as shown in Table 3. These results show that the imperfection sensitivity becomes slightly weak when the temperature is increased and the shell geometric parameter has only a small effect on the imperfection sensitivity. As

ARTICLE IN PRESS Z.-M. Li, H.-S. Shen / International Journal of Mechanical Sciences 50 (2008) 719–731

730

1.2

1.2

1.0

1.0

I: b1 = 0.0 II: b1 = 10.0 I&3

0.6

II&3

I&2

II&2

I&1

0.4

I: b1 = 0.0 II: b1 = 10.0

0.8 q (MPa)

q (MPa)

0.8

II&1

0.4

W*/t = 0.0

0.2

I&3 II&3 I&2 II&2 I&1 II&1

0.6

W*/t = 0.0

0.2

W*/t = 0.1

W*/t = 0.1

0.0

0.0 -4

-2

0

2

4

0

6

2

Δx (mm)

4

6

8

W/t

Fig. 12. Effect of fiber volume fraction on the postbuckling behavior of a braided composite cylindrical shell subjected to hydrostatic pressure combined with axial compression. (a) Load–shortening and (b) load–deflection.

2.5

2.5 1: Z = 100, (m, n) = (1, 5) 2: Z = 200, (m, n) = (1, 4) 3: Z = 500, (m, n) = (1, 3)

2.0

I: b1 = 0.0 II: b1 = 10.0

1: Z = 100, (m, n) = (1, 5) 2: Z = 200, (m, n) = (1, 4) 3: Z = 500, (m, n) = (1, 3)

2.0

I&1

I: b1 = 0.0 II: b1 = 10.0

II&1

1.5

I&1

II&1

II&2 I&2

1.0

I&3

0.5

q (MPa)

q (MPa)

1.5

II&2

1.0

II&3

I&2

I&3 II&3

0.5 W*/t = 0.0

W*/t = 0.0

W*/t = 0.1

W*/t = 0.1

0.0

0.0 -4

0

4 Δx (mm)

8

12

0

2

4

6

8

W/t

Fig. 13. Effect of shell geometric parameter on the postbuckling behavior of braided composite cylindrical shells subjected to hydrostatic pressure combined with axial compression. (a) Load–shortening and (b) load–deflection.

mentioned before for larger imperfection amplitudes, e.g.  W =t40:22, the postbuckling equilibrium path becomes stable, so that no imperfection sensitivity can be predicted. Figs. 10–13 show the postbuckling responses for the braided composite cylindrical shell under a combined loading case (1) analogous to the results of Figs. 5–8 under a combined loading case (2). Now the load-proportional parameter b1 is taken to be 0.0 and 10.0. The results show that an increase in pressure is usually required to obtain an

increase in deformation, and the postbuckling path is stable for both perfect and imperfect shells, and the shell structure is virtually imperfection-insensitive. 4. Concluding remarks A fully nonlinear postbuckling analysis is presented for 3D braided composite cylindrical shells subjected to combined loading of external pressure and axial

ARTICLE IN PRESS Z.-M. Li, H.-S. Shen / International Journal of Mechanical Sciences 50 (2008) 719–731

compression based on a micro–macro-mechanical model. A singular perturbation technique is employed to determine interactive buckling loads and postbuckling equilibrium paths. After comparing the present solutions with the existing ones for some limiting cases available in the literature, the interaction buckling curves and postbuckling response of perfect and imperfect, braided composite cylindrical shells under combined loading cases have been investigated. The results show that the shape of interaction buckling curves depends on the temperature variation and fiber volume fraction, and the postbuckling characteristics additionally depend significantly the load-proportional parameter b2 (or b1). The results reveal that in the combined loading case (2) the postbuckling equilibrium path is unstable and the shell structure is imperfectionsensitive. In contrast, in the combined loading case (1) the postbuckling equilibrium path is stable for both perfect and imperfect cylinders and the shell structure is virtually imperfection-insensitive. Acknowledgment This work is supported in part by the National Natural Science Foundation of China under Grant 50375091. The authors are grateful for the financial support. References [1] Wang X, Dai HL. Thermal buckling for local delamination near the surface of laminated cylindrical shells and delaminated growth. Journal of Thermal Stresses 2003;26:423–42. [2] Wang X, Lu G, Xiao DG. Non-linear thermal buckling for local delamination near the surface of laminated cylindrical shell. International Journal of Mechanical Sciences 2002;44:947–65. [3] Ishikawa T, Chou TW. Elastic behavior of woven hybrid composites. Journal of Composite Materials 1982;16:2–19. [4] Naik RA. Failure analysis of woven and braided fabric reinforced composites. Journal of Composite Materials 1995;29:2334–63. [5] Wang YQ, Wang ASD. Spatial distribution of yarns and mechanical properties in 3D braided tubular composites. Applied Composite Materials 1997;4:121–32. [6] Kwon YW, Cho WM. Multilevel, micromechanical model for thermal analysis of woven-fabric composite materials. Journal of Thermal Stresses 2004;27:59–73. [7] Jensen DW, Pai SP. Influence of local fiber undulation on the global buckling of filament-wound cylinders. Journal of Reinforced Plastics and Composites 1993;12:865–75. [8] Pai SP, Jensen DW. Influence of fiber undulations on buckling of thin filament-wound cylinders in axial compression. Journal of Aerospace Engineering 2001;14:12–20. [9] Elghazouli AY, Chyssanthopoulos MK, Esong IE. Buckling of woven GFRP cylinders under concentric and eccentric compression. Composite Structures 1999;45:13–27. [10] Chyssanthopoulos MK, Elghazouli AY, Esong IE. Validation of FE models for buckling analysis of woven GFRP shells. Composite Structures 2000;49:355–67.

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