Buckling of conical composite shells

Composite Structures 94 (2012) 787–792

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Buckling of conical composite shells F. Shadmehri a,⇑, S.V. Hoa a, M. Hojjati b a Department of Mechanical and Industrial Engineering, Concordia Center for Composites, Center for Research in Polymers and Composites (CREPEC), Concordia University, Montreal, Quebec, Canada H3G1M8 b National Research Council Canada, Montreal, Quebec, Canada

a r t i c l e

i n f o

Article history: Available online 1 October 2011 Keywords: Conical shells Composite structures Buckling analysis

a b s t r a c t A semi-analytical approach is proposed to obtain the linear buckling response of conical composite shells under axial compression load. A first order shear deformation shell theory along with linear strain– displacement relations is assumed. Using the principle of minimum total potential energy, the governing equilibrium equations are found and Ritz method is applied to solve them. Parametric study is performed by finding the effect of cone angle and fiber orientation on the critical buckling load of the conical composite shells. Ó 2011 Published by Elsevier Ltd.

1. Introduction Conical composite shells have wide applications in aerospace/ aeronautical, naval and civil structures. This extensive application of conical composite shells in industry calls for efficient tools to analyze the mechanical behavior of these structures. In many applications the primarily concern is the stability of the structure and analytical solution is necessary to predict the critical buckling load. There have been extensive studies on the buckling of isotropic conical shells under axial load and external pressure [1–8]. Seide [1] proposed a formula for buckling of isotropic conical shell which is independent of boundary conditions and best fits the behavior of long shells. Baruch et al. [8] investigated the stability of simply supported isotropic conical shells under axial load for four different sets of in-plane boundary conditions using Donnell-type theory. However, although laminated composite materials have found extensive industrial applications during the last decades, only few studies have been published targeting the buckling behavior of conical composite shells. Using Donnell-type shell theory, Tong and Wang [9] proposed a power series based solution for buckling analysis of laminated conical shells under axial compressive load and external pressure. Li [10] considered the stability of composite stiffened shell under axial compression load. He assumed classical lamination theory and used Rayleigh–Ritz approximation to solve the governing equations. Sofiyev [11] studied the buckling of orthotropic composite conical shell incorporating the effect of

⇑ Corresponding author. E-mail addresses: [email protected] (F. Shadmehri), hoasuon@alcor. concordia.ca (S.V. Hoa), [email protected] (M. Hojjati). 0263-8223/$ - see front matter Ó 2011 Published by Elsevier Ltd. doi:10.1016/j.compstruct.2011.09.016

thickness variation and time dependent external pressure. Donnell-type shell theory was assumed in his work and Galerkin method and variational technique were applied to obtain the solution. Static, free vibration and buckling analysis of laminated conical shell using finite element method based on higher order shear deformation theory was carried out by Pinto Correia et al. [12]. The effect of variations of the stiffness coefficients on the buckling of laminated conical shells was studied by Goldfeld and Arbocs [13] using classical shell theory and computer code STAGS-A. Patel et al. [14] studied postbuckling characteristics of angle-ply laminated conical shells subjected to torsion, external pressure, axial compression, and thermal loading using the finite element approach. In this paper, axisymmetric and non-axisymmetric formulations for buckling analysis of conical composite shells subjected to axial compression load are developed and semi-analytical solution using Ritz method is obtained. The effect of transverse shear deformation is taken into account since the classical theory of shells was shown not to be accurate enough for moderately thick laminated shells and where the material anisotropy is severe [12,15]. Parametric study is carried out to reveal the influence of the semi-cone angle and the fiber orientation on critical buckling load. 2. Formulation The principle of minimum total potential energy in conjunction with variational techniques is employed to derive the governing equations for general laminated thin-walled conical shells. Linear strain–displacement relation is assumed. Furthermore, using first-order shear deformation shell theory, Kirchhoff hypothesis denoting that the transverse normal to the mid-surface remains perpendicular to it after deformation, is relaxed.

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2.1. Kinematics

2.2. Constitutive law

The coordinate system associated with conical shell is shown in Fig. 1. Based on the shear deformation shell theory assumption, the displacement field can be written as [16].

Assuming orthotropic properties for each layer and neglecting the transverse normal stress, force and moment resultants are related to strains by laminated stiffness coefficients as follows [17]:

Uðx; h; zÞ ¼ uðx; hÞ þ zbx ðx; hÞ Vðx; h; zÞ ¼ v ðx; hÞ þ zbh ðx; hÞ

ð1Þ

Wðx; h; zÞ ¼ wðx; hÞ In these equations u, v and w are the displacements of midsurface and bx and bh are the rotations of a normal to the midsurface about h and x axis respectively. The strain–displacement relations can be stated in terms of mid-surface strain and the curvature of the shell as

ex ¼ e0x þ ze0x eh ¼ e0h þ ze0h cxh ¼ c0xh þ zc0xh cxz ¼ c0xz chz ¼ c0hz

8 9 2 Nx > A11 > > > > > 6A > > > > N h > 12 > > > 6 > = 6 6 A16 xh ¼6 > Mx > > 6 > 6 B11 > > > > 6 > > Mh > > 4 B12 > > > > : ; M xh B16    A44 Qh ¼ Ks Qx A45

A12

A16

B11

B12

A22

A26

B12

B22

A26 B12

A66 B16

B16 D11

B26 D12

B22

B26

D12

D22

B66 D16 ( 0 )

D26

B26 A45 A55

9 38 B16 > e0x > > > > > e0 > > > B26 7 > > 7> h > > > 7> < 0 = 7 B66 c xh 7 > e0x > D16 7 > 7> > 0 > > 7> > eh > > D26 5> > > > : 0 > ; D66 cxh

ð4Þ

chz c0xz

2.3. Total potential energy

ð2Þ The total potential energy of the conical shell subjected to an axial compressive in-plane load consists of the strain energy and the potential energy due to uniaxial compressive force per unit length. It can be written as

where

  @u 1 @v e0h ¼ þ usinðaÞ þ wcosðaÞ @x x sinðaÞ @h   @b 1 @bh x e0x ¼ e0h ¼ þ hx sinðaÞ x sinðaÞ @h @x   @v 1 @u 0 cxh ¼  v sinðaÞ  @h @x x sinðaÞ @w @bh 1 @bx bh 0 0 þ bx cxh ¼ cxz ¼ þ  @x @x x sinðaÞ @h x 1 @w v 0  chz ¼ cosðaÞ þ bh x sinðaÞ @h x sinðaÞ

P¼UþW

e0x ¼

a in above equations is semi-cone angle.

ð5Þ

in which U represents the strain energy, while W is the work of the applied in-plane compressive force and are defined as

ð3Þ

U¼

1 2

ZZ

 xh

Nx e0x þ Nh e0h þ Nxh c0xh þ Mx e0x þ M h e0h þ Mxh c0xh

1 þQ y c0xz þ Q x c0hz rdh dx W ¼ 2

ZZ

 2 b xx @w rdh dx N @x xh

ð6Þ

The principle of minimum total potential energy which can give the governing equations of the conical shell subjected to compressive axial load can be stated as

dP ¼ dU þ dW ¼ 0

ð7Þ

where dU and dW are first variation of strain energy and work of the applied in-plane compressive force, respectively. 2.4. Ritz method Ritz method is a powerful tool to determine approximate solutions of the governing equations by using the principle of minimum total potential energy (Eq. (7)). This method has an advantage of giving directly the solution from variational statement by bypassing the equilibrium equations [18]. The approximate solutions for displacements are assumed in series form as

½uðx; hÞ; v ðx; hÞ; wðx; hÞ; bx ðx; hÞ; bh ðx; hÞ m X n h i X ¼ Uuij ; Uvij ; Uwij ; Ubijx ; Ubijh i¼1

ð8Þ

j¼1

where Uuij ; Uvij ; Uwij ; Ubijx and Ubijh are approximation functions and they need to satisfy convergence and completeness requirements [18]. Considering Eq. (8) and using Eqs. (2)–(6) and rewriting them in a matrix form, one can derive the first variation of the strain energy and work of the applied force by performing repeated integration by part as

dU ¼

Fig. 1. Conical shell coordinate system.

ZZ

dW s ¼

d½CT ½UT ½BT ½F½B½U½Crdh dx

ZZxh

ð9Þ ^ U½Crdh dx d½CT ½UT ½B½ xh

F. Shadmehri et al. / Composite Structures 94 (2012) 787–792

b matrices are defined in Appendix in Eq. (9), [C], [U], [B], [F] and ½ B A. Substituting Eq. (9) into Eq. (7) and rearranging the matrices, one can get

d½CT

ZZ

 b > ½UÞrdh dx ½C ¼ 0 ð½UT < ½BT ½F½B  ½ B

ð10Þ

Since Eq. (10) is required to be satisfied for all possible values of the coefficient vector [C], it implies that the coefficient of variation d[C]T must vanish as

ð11Þ

where

½Z ¼

ZZ

b > ½UÞrdh dx ð½UT < ½BT ½F½B  ½ B

 ð12Þ

xh

Eq. (11) is an eigenvalue problem and the corresponding lowest eigenvalue is the critical buckling load. The buckling load is the existence of a non-trivial solution of Eq. (11) which means the determinant of matrix [Z] must be vanished, that is

detjZj ¼ 0

ð13Þ

2.5. Axisymmetric problem Up to this point, it was assumed that the buckling mode is not axisymmetric. In this section we will consider the axisymmteric mode in which v, the displacements of mid-surface in circumferential direction and bh, the rotations of a normal to the mid-surface about x axis is equated to zero. Consequently, the displacement field, Eq. (1), reduces to

Uðx; zÞ ¼ uðxÞ þ zbx ðxÞ

0 x 0 h

ex ¼ e þ ze eh ¼ e þ ze cxz ¼ c0xz

ð15Þ

where 1 e0h ¼ x sinð aÞ ðusinðaÞ þ wcosðaÞÞ

e0x ¼ dbdxx e0h ¼ 1x ðhx Þ c0xz ¼ dw þ bx dx

ð16Þ

The force and moment resultants, Eq. (4), reduce to

8 9 2 Nx > A11 > > > > > > > 6 > > A12 N > > < h= 6 6 6 Mx ¼ 6 B11 > > > > 6 > > Mh > > 4 B12 > > > > : ; 0 Qx

 x  xt cosðjhÞ xb  xt   x  xt sinðjhÞ Uvij ¼ cos ip xb  xt   x  xt sinðjhÞ Uwij ¼ sin ip xb  xt   x  xt sinðjhÞ Ubijx ¼ cos ip xb  xt   x  xt cosðjhÞ Ubijh ¼ sin ip xb  xt

ð18Þ

where xt and xb are the top and bottom coordinate of the cone shown in Fig. 1. For axisymmetric buckling problem the approximation functions are only function of axial coordinate x and therefore get the form of



 x  xt xb  xt   x  xt w Ui ¼ sin ip xb  xt   x  xt bx Ui ¼ cos ip xb  xt

Uui ¼ sin ip

ð19Þ

The buckling load for each conical shell configuration (different geometry or lay-up sequence) is obtained by solving Eq. (13) to find b xx . This procedure should be repeated based the minimum value of N on the axisymmetric assumption as well and finally whichever value b xx that is the lowest would be selected as the critical buckling of N load. 4. Validation

and the strain–displacement relations, Eq. (3), get simplified accordingly as

e0x ¼ du dx



ð14Þ

Wðx; zÞ ¼ wðxÞ

0 x 0 h

S2-type simply supported boundary condition by seeking solutions of the form

Uuij ¼ sin ip

xh

½Z½C ¼ 0

789

A12

B11

B12

A22

B12

B22

B12

D11

D12

B22

D12

D22

0

0

0

38 0 9 e > > > > x0 > > 7> > 0 7> e > > > 7< h0 = 7 0 7 ex > 0 > 7> > 0 5> > > eh > > > > : 0 > ; A55 cxz 0

Analytical solution for buckling loads of cross-ply circular cylindrical shells under axial load was presented by Khdeir et al. [19] and Shadmehri et al. [20]. Since exact solution is available for cross-ply cylindrical shells, it is selected for validation purposes. In order to verify with a composite cylindrical shell, the cone angle is set to zero. The approximation functions are selected to satisfy the simply-supported and clamped boundary conditions defined in [19,20] and material properties and geometric specifications are set accordingly (see [19,20] for details). The critical buckling loads for two different cases are compared with references [19,20] and shown in Table 1. 5. Numerical results

ð17Þ

By applying the principle of minimum total potential energy, equation similar to Eq. (10) will be obtained, which is not presented here for the sake of saving space. The definitions of the involved matrices are different from Eq. (10) and are presented for axisymmetric problem in Appendix B. 3. Solution Following the Ritz method, the boundary conditions can be applied through the assumed displacement functions. These approximation functions must satisfy the essential boundary conditions and must be linearly independent and complete. In this paper, we consider the conical composite shells subjected to

In order to evaluate the effect of semi-cone angle and fiber orientation on the critical buckling load of the conical composite shells, typical properties of graphite-epoxy material are assumed as shown in Table 2. 5.1. Effect of semi-cone angle on critical buckling load r A set of conical shells with the radius tothickness ratio of 100 rb b ¼ 100 and radius to length ratio of 10 ¼ 10 are assumed. H L The base radius (rb) is kept constant while the semi-cone angle (a) varies from 0 to 44° to evaluated the effect of a on dimensionless critical buckling load defined as

N¼

b xx L2 N H3 E22

ð20Þ

b xx is the critical buckling load, H is the total thickness of the where N laminate, and L is the length of the conical shell shown in Fig. 1.

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  Table 1 N xx L2 b0 ¼ b ; HR ¼ 10; RL ¼ 1 .a Validation of dimensionless critical buckling load of cross-ply cylindrical shell N 100h3 E 22

a

Lamination

SS

CC

Present work

Ref. [20]

Ref. [19]

Present work

Ref. [20]

Ref. [19]

0/90/0 0/90

0.2765 0.1525

0.2765 0.1525

0.2813 0.1670

0.4168 0.2406

0.4168 0.2406

0.4197 0.2508

R: cylinder radius, L: cylinder length, H: total thickness of the cylinder.

Table 2 Material properties of conical shell for buckling analysis. Material properties E11 = 3.05  107 psi E22 = E33 = E1/40 G12 = G13 = 0.6E22 G23 = 0.5E22 m12 = 0.25

The critical buckling load is obtained by calculating nonaxisymmetric and axisymmetric buckling loads and selecting the lowest value. Three lay-ups (two cross-ply lay-ups and one transversely isotropic lay-up) are studied and results are shown in Fig. 2. It was found that for all three lay-ups the axisymmetric buckling loads are always lower than the non-axisymmetric ones. For [0/90] the difference is about 17% while for [0/90/0] and [0/ +45/45/90]s the difference is about 0.5%. As can be seen from Fig. 2 the critical buckling load decreases as the semi-cone angle (a) increases for all three lay-ups. This decrease is not at constant rate. In the range of a between 0° and 20° the reduction in critical buckling load is not considerable (about 7%). After a exceeds 20° the reduction in critical buckling load is considerable (about 33% from initial value at a = 0). The reduction in buckling strength can be attributed to the change in the geometry of the conical shells. As a increases the radius of the small side of the conical shell gets smaller and consequently the critical buckling load decreases. From lay-up sequence point of view, [0/90/0] lay-up has the highest critical buckling load for all a values, offering that 0° layers laid at the outer skin of the conical shells provide more stiffness than other two lay-ups.

α (deg.) Fig. 2. Dimensionless critical buckling load ðNÞ vs. semi-cone angle (a).

Axisymmetric solution Non-axisymmetric solution

5.2. Effect of fiber orientation on critical buckling load Effect of fiber orientation on critical buckling load of conical shells with geometrical specification of rHb ¼ 100; rLb ¼ 10 and for four different semi-cone angles are shown in Figs. 3 and 4. The angle-ply laminated conical shells with [+h/h] lay-up are considered for analysis. Fiber orientation h is defined with respect to x-axis (Fig. 1) and alternates between 0° and 90°. For comparison purposes, the axisymmetric and non-axisymmetric buckling loads are plotted in Fig. 3 for semi-cone angle a = 10°. It can be seen that non-axisymmteric buckling load is less than the axisymmteric one for all fiber angles except at 0° and from 85° to 90° where both axisymmetric and non-axisymmetric buckling loads are almost the same. So one can say that non-axisymetric buckling load is representative of critical buckling load in this case. This trend is the same for other angle-ply cones with the same rHb and rLb ratios and different semi-cone angle and the results are shown in Fig. 4. The curves shown in Fig. 4 indicate a rapid drop in the buckling strength of angle-ply conical shells as the fiber angle (h) varies from 0° up to 45° (about 77% reduction). Then one can observe a relatively constant period where h alternates between 45° and 90° with a minimum around 55° and maximum around 70°. Also, as can be expected from the results of previous section, it is clear

θ (deg.) Fig. 3. Axisymmetric and non-axisymmetric dimensionless buckling load ðNÞ vs. fiber angle (h) for a = 10°.

α=0 α = 10 α = 20 α = 30

θ (deg.) Fig. 4. Dimensionless critical buckling load ðNÞ vs. fiber angle (h).

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from Fig. 4 that increasing the semi-cone angle a has detrimental effect on critical buckling load. 6. Conclusions Buckling analysis of laminated conical shells has been performed through a shear deformation shell theory. The principle of minimum potential energy along with the Ritz method has been used to obtain the governing equation and to find the solution for buckling problem. Both non-axisymmetric and axisymmetric formulation have been derived and solved for each laminated conical shell and the lowest buckling load has been selected as critical buckling load. The following conclusion can be drawn from parametric studies presented in the numerical results: 1. For thin and short conical shells considered in this study the critical buckling load decreases with increasing semi-cone angle. The reduction in critical buckling load becomes more pronounced as the semi-cone angle exceeds 20°. This can be important from design point of view. 2. The critical buckling load decreases with increasing fiber orientation of angle-ply thin and short conical shells.

Acknowledgments Authors of this paper would like to acknowledge the support of this research by CRIAQ (Consortium for Research and Innovation in Aerospace in Quebec), NSERC (Natural Sciences and Engineering Research Council of Canada), Bell Helicopter Canada, Bombardier Inc., and Dema Aerospace. Appendix A The matrices appearing in Eq. (9) are:

½CT ¼ cu11

cv11

cb11x

cw 11

cb11h

u U11 0 0 0 0 0 Uv 0 0 0 11 0 Uw 0 0 ½U ¼ 0 11 bx 0 0 U11 0 0 0 0 0 0 Ub11h

cvmn

   cumn

... Uu21 .. . Uumn v

... U21 .. . ... Uw 21 .. . ... Ub21x .. . ... U

bh 21

.. .

cw mn

0

x cbmn

0

0

0

v

Umn

0

0

0

0

Uwmn

0

0

0

0

x Ubmn

0

0

0

0

h cbmn



0 0 0 0 Ubh

0 0 b ¼ 0 ½ B 0 0

0

0

0 b xx N

0

0

0

@2 @x2

sinðaÞ @ @x r

þ

0 0 0 0 0



0 0

0

0

0

0

0

0

r ¼ x sinðaÞ Appendix B The matrices related to axisymmetric problem are:

h i bx u w bx ½CT ¼ cu1 cw 1 c 1    c m cm c m u U 11 ½U ¼ 0 0

0

0

   Uu21

   Uumn

Uw11

0

   Uw 21



0

0 Uw mn

bx 21



0

0

0

@ @x 1 r sinðaÞ 0 ½B ¼ 0 0 A11 A 12 ½F ¼ B11 B12 0 0 b ¼ 0 ½ B 0

bx 11

U

 U

0 1 r

cosðaÞ 1 r

0 @ @x

A12

B11

B12

A22

B12

B22

B12

D11

D12

B22

D12

D22

0

0

0

b xx N

@2 @x2

0 þ sinðr aÞ 0

0

0

0 0 Ubx mn

0 @ @x sinðaÞ 1 0

0

0

0 0 0 0 A55

 0 @ 0 @x 0

r ¼ x sinðaÞ

mn

References

@ @x 1 sinðaÞ r 1 @ r @h 0 ½B ¼ 0 0 0 0 A11 A 12 A16 B 11 ½F ¼ B12 B 16 0 0

0 @ @x



0

1 @ r @h 1 sinð r

1 cosð r

aÞ

aÞ

0

0

0

0

0

1 r

0

0

 1r cosðaÞ

1 @ r @h @ @x

0

A12 A22

A16 A26

B11 B12

A26

A66

B16

A12

B16

D11

A22

B26

D12

B26

B66

0 0

A12 A22

0 0 0 0 0 0 @ 0 @x @ sinðaÞ 1r @h 1 @ @  1x r @h @x 0 1 1 0

B16 B26

0 0

A26

B66

0

D12

D16

0

D22

D26

0

D16

D26

D66

0

0

0

0

0

A45

0

0

0

0

A45

0 0 0 0 A45 A 0 0

55

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