The effect of elastic foundations on the nonlinear buckling behavior of axially compressed heterogeneous orthotropic truncated conical shells

Thin-Walled Structures 80 (2014) 178–191

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The effect of elastic foundations on the nonlinear buckling behavior of axially compressed heterogeneous orthotropic truncated conical shells A.H. Sofiyev n Department of Civil Engineering of Suleyman Demirel University, 32260 Isparta, Turkey

art ic l e i nf o

a b s t r a c t

Article history: Received 12 February 2014 Received in revised form 24 March 2014 Accepted 24 March 2014 Available online 14 April 2014

The buckling problem of a heterogeneous orthotropic truncated conical shell subjected to an axial load and surrounded by elastic media is analyzed based on the finite deformation theory. Using von-Karman nonlinearity, the governing equations of elastic buckling of heterogeneous orthotropic truncated conical shells surrounded by elastic media are derived. The governing equations are solved using superposition and Galerkin methods and obtained expressions for upper and lower critical axial loads. The influences of elastic foundations, heterogeneity, orthotropy and geometric characteristics on the upper and lower critical loads of conical shells with and without elastic foundations are studied in detail. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear buckling Orthotropic materials Heterogeneity Elastic foundations Critical lower loads

1. Introduction The composite truncated conical shell, as an important structural component, has extensive applications in many engineering fields. Deformation of such structures under an axial load causes the phenomenon of snap-through buckling. The evaluation of such buckling behavior and critical load becomes an important topic in research. A lot of work on the buckling analysis has been carried out for homogeneous truncated conical shells under an axial compressive load [1–8]. As flexible structures, truncated conical shells are sensitive to the heterogeneity. The effect of heterogeneity on load carrying capability is thus important in the buckling analysis of shell structures [9–29]. In addition, the effects of elastic foundations supporting the shells on the buckling behavior of structures are important. The structure–foundation interactions are frequently considered using Winkler and Pasternak elastic foundations. As a result, many studies have been carried out on the analysis and behavior of shells on elastic foundations, including a significant number of studies on the large deflection and buckling behavior of homogeneous isotropic and orthotropic shells surrounded by an elastic medium. Gazizov and Zaynashev [30] investigated the stability of truncated conical shells on an elastic foundation beyond the elasticity limit. Bajenov [31] presented the solutions

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http://dx.doi.org/10.1016/j.tws.2014.03.016 0263-8231/& 2014 Elsevier Ltd. All rights reserved.

of vibration, buckling and bending problems of cylindrical shells in the elastic and visco-elastic medium. Sun and Huang [32] investigated the general bending problems of conical shells under the arbitrary loads and boundary conditions and surrounded by the elastic foundation and the problem is reduced to find the displacement function and the general solution of the eight-order differential equation is obtained in series form. Nath and Jain [33] analyzed the nonlinear axisymmetric static buckling of cylindrically orthotropic full and annular shallow spherical shells continuously supported on Winkler and Pasternak elastic foundations. Luo and Teng [34] developed a finite element formulation for the buckling analysis of shells of revolution on nonlinear elastic foundations. Naili and Oddou [35] analyzed theoretically the buckling of short cylindrical shell surrounded by an elastic medium on the basis of the asymptotic method. Nie et al. [36] presented an asymptotic solution for nonlinear buckling of elastically restrained imperfect, orthotropic shells on elastic foundation using iteration method. Kurpa et al. [37] investigated solution the nonlinear bending problems for orthotropic shallow shells on an elastic foundation using the R-function method. Shen [38] analyzed postbuckling behavior of axially-loaded laminated cylindrical shells surrounded by an elastic medium. PanahandehShahraki et al. [39] investigated the nonlinear buckling of laminated composite curved panels constrained by a Winkler tensionless foundation. Shams et al. [40] analyzed nonlinear buckling of a single spherical shell imperfectly bonded to an infinite elastic matrix under a compressive remote load, using Galerkin and modified Newton–Raphson methods.

A.H. Sofiyev / Thin-Walled Structures 80 (2014) 178–191

There have been relatively few studies on the buckling behavior of heterogeneous isotropic and orthotropic truncated conical shells surrounded by elastic media. Sofiyev [41] examined the linear buckling of FGM truncated conical shells subjected to an axial compressive load and surrounded by Winkler–Pasternak foundations. Bagherizadeh et al. [42] investigated the linear mechanical buckling of functionally graded material cylindrical shells surrounded by Pasternak elastic foundation. Kiani et al. [43] investigated static and dynamic analysis of an FGM doubly curved panel surrounded by the Pasternak elastic foundation in the linear formulation. Sofiyev et al. [44] analyzed the effect of two-parameter elastic foundation on the upper values of critical parameters for non-homogeneous orthotropic shells. Shen and Wang [45] presented the thermal buckling and postbuckling behaviors of fiber reinforced composite (FRC) laminated cylindrical shells embedded in a large outer elastic medium and subjected to a uniform temperature rise, based on a higher order shear deformation shell theory that includes shell-foundation interaction. Dung et al. [46] solved the linear stability problems of the FG truncated conical shell with FG stiffeners and surrounded by an elastic medium using partial differential equations in terms of three displacement components. The above mentioned problems are solved in a linear formulation. It is possible for heterogeneous truncated conical shell structures surrounded by elastic media to induce large deformations under the loading. Therefore, it is very important to develop an accurate, reliable analysis towards the understanding of the nonlinear buckling characteristics of heterogeneous orthotropic truncated conical shells surrounded by an elastic medium. In this paper the nonlinear buckling response of orthotropic truncated conical shells with variable material properties and surrounded by elastic media is investigated. The governing equations are derived in terms of the Airy stress function and deflection for the nonlinear buckling analysis. The model which describes the foundation includes Winkler (normal) and Pasternak (shear) foundation parameters. The governing equations are solved using superposition and Galerkin methods. The results of numerical calculations are presented in tables and figures, which show the influences of elastic foundations, heterogeneity, orthotropy and geometric parameters on the values of the upper and lower critical loads of the truncated conical shell.

2. Formulation of the problem Consider a truncated conical shell subjected to an axial load ðT ax Þ and surrounded by a two-parameter elastic medium, as shown in Fig. 1. The Pasternak model is used to describe the reaction of an elastic foundation on a truncated conical shell.

179

R1 and R2 are the average radii of the cone at its small and large edges, h is the cone thickness, γ is the semi-vertex angle of the cone and L is the truncated cone length along its generator. S1 is the distance of the smaller end of the truncated conical shell from the vertex. The truncated conical shell is referred to an orthogonal curvilinear coordinates S; θ; ζ as shown in Fig. 1, in which S is in the generatrix direction measured from the vertex of the shell, θ is in the circumferential direction, and ζ is in the normal direction measured from the reference surface, positive inward. The corresponding displacements at the mid-surface are designated u, v and w in S; θ and ζ directions, respectively. The load–displacement relationship of the foundation is assumed to be
2  ∂ w 1 ∂w 1 ∂2 w NðS; θÞ ¼ K w w  K p þ þ ð1Þ ∂S2 S ∂S S2 ∂θ1 2 where NðS; θÞ is the force per unit area, K w (in N=m3 ) is the Winkler foundation stiffness and K p (in N/m) is the shearing layer stiffness of the foundation, and θ1 ¼ θ sin γ [32].

3. Governing relations The geometrical relations between the strain and displacement components for truncated conical shells, according to von-Karman nonlinear theory are given as follows [3,21]:

 ∂u 1 ∂w 2 1 ∂v u w cot γ 1 ∂w 2 þ  þ 2 eS ¼ þ ; eθ ¼ ∂S 2 ∂S S ∂φ S S 2S ∂θ1
 1 ∂u ∂v v 1 ∂w ∂w ð2Þ 2eSθ ¼ þ  þ S ∂θ1 ∂S S S ∂S ∂θ1 where eS ; eθ ; eSθ are the strain components on the reference surface. The stress–strain relations for heterogeneous orthotropic truncated conical shells, according to Hooke’s law are

sS ¼ Q 11 eS þ Q 12 eθ þ ζðQ 11 χ S þ Q 12 χ θ Þ sθ ¼ Q 21 eS þ Q 22 eθ þ ζðQ 21 χ S þ Q 22 χ θ Þ sSθ ¼ Q 66 eSθ þ ζQ 66 χ Sθ

ð3Þ

where χ S and χ θ are curvatures of the deformed conical shell in the S and θ directions, wherein χ Sθ is the torsion of the reference surface and expressed as [1–4] χS ¼ 

∂2 w ∂S

2

;

χθ ¼ 

1 ∂2 w 1 ∂w ;  S2 ∂θ21 S ∂S

χ Sθ ¼ 

1 ∂2 w 1 ∂w þ S ∂S∂θ1 S2 ∂θ1

ð4Þ

and Q ij ði; j ¼ 1; 2; 6Þ are the quantities and the following definitions apply: Q 11 ¼

E0S φðζÞ ; 1  νSθ νθS

Q 22 ¼

E0θ φðζÞ ; 1  νSθ νθS

Q 12 ¼ Q 21 ¼ νθS Q 11 ¼ νSθ Q 22 ;

Fig. 1. Truncated conical shell subjected to the axial compressive load and surrounded by an elastic medium.

180

A.H. Sofiyev / Thin-Walled Structures 80 (2014) 178–191

Q 66 ¼ 2G0 φðζÞ;

ζ ¼ ζ=h

ð5Þ

in which E0S and E0θ are Young’s moduli in the S and θ directions, respectively, G0 is the shear modulus on the plane of homogeneous orthotropic material of shell; φðζÞ is a continuous function giving the variations of Young’s moduli and shear modulus of orthotropic materials; νSθ and νθS are Poisson’s ratios, assumed to be constant and satisfying νθS E0S ¼ νSθ E0θ . The force and moment resultants are defined by the following integral equations [47]: Z h=2 ½ðN S ; N θ ; NSθ Þ; ðM S ; M θ ; M Sθ Þ ¼ ðsS ; sθ ; sSθ Þ½1; ζ dζ ð6Þ  h=2

The relations between the forces and Airy stress function, Ψ , are [3] NS ¼

1 ∂2 Ψ 1 ∂Ψ ; þ S2 ∂θ21 S ∂S

Nθ ¼

∂2 Ψ ∂S

2

;

NSθ ¼ 

1 ∂2 Ψ 1 ∂Ψ þ S ∂S∂θ1 S2 ∂θ1

ð7Þ

Introduction of relations (4) into (3), and then substituting the resulting expressions in (6) together with (7), after mathematical operations yields the following constitutive expressions for strains on the reference surface and moments: ! ! 1 ∂2 Ψ 1 ∂Ψ ∂2 Ψ ∂2 w 1 ∂2 w 1 ∂w eS ¼ b11 2 2 þ þ b12 2  b13 2  b14 2 2 þ S ∂θ1 S ∂S ∂S ∂S S ∂θ1 S ∂S ! ! 2 2 2 1 ∂ Ψ 1 ∂Ψ ∂ Ψ ∂ w 1 ∂2 w 1 ∂w eθ ¼ b21 2 2 þ þ b22 2  b23 2  b24 2 2 þ S ∂θ1 S ∂S ∂S ∂S S ∂θ1 S ∂S

 1 ∂Ψ 1 ∂2 Ψ 1 ∂w 1 ∂2 w þ b32 2 ð8Þ   eSθ ¼ b31 2 S ∂θ1 S ∂S∂θ1 S ∂θ1 S ∂S∂θ1 M S ¼ c11

1 ∂2 Ψ 1 ∂Ψ þ S2 ∂θ21 S ∂S

! þc12

∂2 Ψ 2

 c13

∂2 w 2

 c14

1 ∂2 w 1 ∂w þ S2 ∂θ21 S ∂S

!

∂S ∂S ! ! 1 ∂2 Ψ 1 ∂Ψ ∂2 Ψ ∂2 w 1 ∂2 w 1 ∂w M θ ¼ c21 2 2 þ þc22 2  c23 2  c24 2 2 þ S ∂θ1 S ∂S ∂S ∂S S ∂θ1 S ∂S

 2 2 1 ∂Ψ 1 ∂ Ψ 1 ∂w 1 ∂ w   þ c32 2 ð9Þ M Sθ ¼ c31 2 S ∂θ1 S ∂S∂θ1 S ∂θ1 S ∂S∂θ1

where the following definitions are applied: c11 ¼ a111 b11 þ a112 b21 ;

c12 ¼ a111 b12 þ a112 b22 ; 1 1 c13 ¼ a11 b13 þ a12 b23 þ a211 ; c14 ¼ a111 b14 þ a112 b24 þ a212 ; c21 ¼ a121 b11 þ a122 b21 ; c22 ¼ a121 b12 þ a122 b22 ; c23 ¼ a121 b13 þ a122 b14 þ a221 ; c24 ¼ a121 b14 þ a122 b13 þ a222 ; c31 ¼ a166 b31 ; c32 ¼ a166 b32 þ a266 ; b11 ¼ a022 =L0 ; b12 ¼  a012 =L0 ; b13 ¼ ða012 a121  a111 a022 Þ=L0 ; b14 ¼ ða012 a122  a112 a022 Þ=L0 ; b21 ¼  a021 =L0 ; b22 ¼ a011 =L0 ; b23 ¼ ða021 a111  a121 a011 Þ=L0 ; b24 ¼ ða021 a112 a122 a011 Þ=L0 ; b31 ¼ 1=a066 ; b32 ¼  a166 =a066 ; L0 ¼ a011 a022  a012 a021 ; k þ 1 Z 1=2 k þ 1 Z 1=2 E0S h E0θ h k k ζ φðζÞ dζ; ak22 ¼ ζ φðζÞ ak11 ¼ 1  νSθ νθS  1=2 1  νSθ νθS  1=2 ak12 ¼ νθS ak11 ¼ ak21 ¼ νSθ ak22 ; Z 1=2 k kþ1 ζ φðζÞ dζ k ¼ 0; 1; 2: ak66 ¼ 2G0 h

follows [3,32]: ∂N S ∂N Sθ NS  N θ þ ¼0 þ ∂S ∂θ1 S

ð11Þ

∂N Sθ 1 ∂N θ 2N Sθ þ ¼0 þ S ∂θ1 ∂S S

ð12Þ

∂2 M S ∂S

2

þ

2 ∂M S 2 ∂2 M Sθ 1 ∂M θ 2 ∂M Sθ þ þ  S ∂S S ∂S∂θ1 S ∂S S2 ∂θ1

1 ∂2 M θ cot γ ∂2 w Nθ þ NS 2 þ S S2 ∂θ1 2 ∂S
2 
 N θ 1 ∂ w ∂w 1 ∂2 w 1 ∂w þ 2NSθ þ þ  2 2 ∂S S ∂S∂θ1 S ∂θ1 S S ∂θ1 ! 2 ∂ w 1 ∂w 1 ∂2 w þ þ Kww  Kp þ ¼0 ∂S2 S ∂S S2 ∂θ21 þ

cot γ ∂2 w 2 ∂2 eSθ 2 ∂eSθ ∂2 eθ 1 ∂2 eS   þ þ S ∂S2 S ∂S∂θ1 S2 ∂θ1 ∂S2 S2 ∂θ1 2
 2 ∂eθ 1 ∂eS 1 ∂w 2 2 ∂w ∂2 w   4 þ þ 3 S ∂S S ∂S S ∂θ1 S ∂θ1 ∂S∂θ1 " #
 2 1 ∂2 w ∂2 w ∂2 w 1 ∂w ∂2 w þ 2 þ ¼0 þ 2 2 ∂S∂θ1 S ∂S ∂S2 S ∂S ∂θ1

ð10Þ

 1=2

4. Governing equations Using a flexible shell theory, which contains relations vonKarman-Donnell-type of kinematic nonlinearity, the nonlinear stability and strain compatibility equations of truncated conical shells surrounded by two-parameter elastic media is expressed as

ð14Þ

Introduction of relations (7) into Eqs. (11) and (12), these two equations are satisfied identically. Therefore, the following governing equations are used for the stability of truncated conical shells surrounded by elastic media: ∂2 M S ∂S

2

þ

2 ∂M S 2 ∂2 M Sθ 1 ∂M θ 2 ∂M Sθ 1 ∂2 M θ cot γ Nθ þ þ  þ þ S ∂S S ∂S∂θ1 S ∂S S2 ∂θ1 S2 ∂θ1 2 S



 Nθ 1 ∂2 w ∂w 1 ∂2 w 1 ∂w þ 2N þ  Sθ S ∂S∂θ1 S2 ∂θ1 S S ∂θ1 2 ∂S ∂S2 ! ∂2 w 1 ∂w 1 ∂2 w þ þ Kww  Kp þ ¼0 ∂S2 S ∂S S2 ∂θ21 þ NS

∂2 w

þ

cot γ ∂2 w 2 ∂2 eSθ 2 ∂eSθ ∂2 eθ 1 ∂2 eS   þ þ S ∂S2 S ∂S∂θ1 S2 ∂θ1 ∂S2 S2 ∂θ1 2
2 2 ∂eθ 1 ∂eS 1 ∂w 2 ∂w ∂2 w   4 þ þ 3 S ∂S S ∂S S ∂θ1 S ∂θ1 ∂S∂θ1 "
2 2 # 1 2 ∂2 w ∂ w 1 ∂w ∂2 w þ 2 ∂∂Sw2 ∂θ ¼0 þ 2  1 ∂S∂θ1 S ∂S ∂S2 S

dζ;

ð13Þ

ð15Þ

ð16Þ

Introduction of relations (8) and (9) in Eqs. (15) and (16), together with Eq. (7), then considering the variable z ¼ lnðS=S1 Þ and Ψ 1 ¼ Ψ e  2z is taken into account instead of Ψ for convenience of mathematical operations in a later stage, the set of governing equations for Ψ 1 and w can be written in the following form: L11 ðΨ 1 Þ þ L12 ðwÞ þL13 ðΨ 1 ; wÞ ¼ 0

ð17Þ

L21 ðΨ 1 Þ þ L22 ðwÞ þL23 ðw; wÞ ¼ 0

ð18Þ

where Lij ði ¼ 1; 2; 3Þ are differential operators and all symbols used in Eqs. (17) and (18) are described in detail in Appendix A. Eqs. (17) and (18) are the governing equations for heterogeneous orthotropic circular truncated conical shells on Winkler– Pasternak elastic foundations. For derivation of Ψ 1 and w, Eqs. (17) and (18) should be solved simultaneously.

A.H. Sofiyev / Thin-Walled Structures 80 (2014) 178–191

ϑ17 ¼  0:5βm ðβ2n þ 3β2m  1:5Þ;

5. Solution of governing equations The boundary conditions of a truncated conical shell are assumed as simply supported: w¼0

as z ¼ 0 and z ¼ z0

ð19Þ

where the following definition applies: z0 ¼ lnð1 þ L=S1 Þ: The solution of governing equations is sought in the following form [3,19]: w ¼ f 1 ez sin ðβm zÞ sin ðβn θ1 Þ þ f 2 sin 2 ðβm zÞ

ð20Þ

where f 1 and f 2 are amplitudes, βm ¼ mπ=z0 ; βn ¼ n= sin γ in which m is number of longitudinal half-waves and n is number of circumferential waves. In addition, the first term corresponds to the small displacements in the theory of stability and the second term corresponds to the inward bulging in the large displacement of the conical shell. Substituting (20) in the right-hand side of Eq. (18), and solving the resulting differential equation, applying the superposition method, the following expression is obtained for Ψ 1 : Ψ 1 ¼ A1 f 1 e  z sin ðβm zÞ sin ðβn θ1 Þ þ A2 f 1 e  z cos ðβm zÞ sin ðβn θ1 Þ þ A3 f 2 e  z cos ð2βm zÞ þ A4 f 2 e  z sin ð2βm zÞ 2

2

2

ϑ19 ¼ 2:5β2m ;

2

ϑ22 ¼ 0:5b14 :

η1 ¼ b22 β4m þ b11 β4n þ β2m ð6b21  6b12  23b22 þ b11 Þ β2n ð8b31 þ4b12 þ 4b21 þ 2b11 Þ þð2b31 þ b21 þb12 Þβ2m β2n þ 12b22 þ 6b12  6b21  3b11 η2 ¼ β3m ðb21  b12  8b22 Þ  βm β2n ð8b31 þ 5b12 þ 3b21 Þ þβm ð28b22  4b11 11b21 þ 11b12 Þ; η3 ¼ 16b22 β4m þ 4β2m ð6b21 6b12  23b22 þ b11 Þ þ12b22 þ 6b12  6b21  3b11 ; η4 ¼  8β3m ðb21 b12  8b22 Þ þ2βm ð11b21  11b12  28b22 þ 4b11 Þ; η5 ¼ 16b22 β4m þ 4ð3b21 þ b11  5b22  3b12 Þβ2m ; η6 ¼  8ðb21  b12  4b22 Þβ3m þ4ðb11 þ b21 b12  b22 Þβm ; η7 ¼ 16b22 β4m þ 4ð3b21 þ b11  5b22  3b12 Þβ2m þ16ð2b31 þ b21 þ b12 Þβ2m β2n  8ðb31 þ b21 þb11 Þβ2n ; η8 ¼  8ðb21  b12  4b22 Þβ3m þ4ðb11 þ b21 b12  b22 Þβm η9 ¼ b22 β4m þ ð3b21 þ b11 5b22  3b12 Þβ2m þ ð2b31 þ b21 þ b12 Þβ2m β2n 2ðb31 þ b21 þb11 Þβ2n þ b11 β4n ; η10 ¼ ðb21  b12  4b22 Þβ3m þ 2ðb11 þ b21  b12  b22 Þβm

2

þ A8 f 1 sin ð2βm zÞ cos ð2βn θ1 Þ þ ðA91 f 1 f 2 þA92 f 1 Þ cos ðβm zÞ sin ðβn θ1 Þ   þ A101 f 1 f 2 þ A102 f 1 sin ðβm zÞ sin ðβn θ1 Þ

þð4b31 þ 3b12 þ b21 Þβm β2n ; η11 ¼ 81b22 β4m  9ð3b12  3b21  b11 þ 5b22 Þβ2m

þ A11 f 1 f 2 cos ð3βm zÞ sin ðβn θ1 Þ þA12 f 1 f 2 sin ð3βm zÞ sin ðβn θ1 Þ 2

þ A13 f 1 f 2 cos ð4βm zÞ þA14 f 1 f 2 sin ð4βm zÞ þ A15 f 1 cos ð2βn θ1 Þ þ A16 f 2 e  z  ½1 þe2z þ cos ðβn θ1 ÞTS21 =2

ϑ17 η11  ϑ18 η12 ϑ17 η12 þ ϑ18 η11 ; A12 ¼ ; η211 þ η212 η211 þ η212

A13 ¼

ð22Þ

In above mentioned definitions, the following notations are introduced: ϑ1 ¼ b14 ½ðβ2n  1Þ2 þ β2m  þβ2m ½ð  2b32 þ b13 þ b24 Þβ2n þ b23 ðβ2m þ 1Þ; ϑ2 ¼ ðb24  b13 Þβm ½β2n β2m 1; ϑ3 ¼  ½8b23 β4m þ 2ðb23 þ b14 Þβ2m þ 0:5b14 ; ϑ4 ¼ βm ½4ðb13  b24 Þβ2m  ðb13 þb24 Þ; ϑ5 ¼ β2m ; ϑ6 ¼ 0:25β2m β2n  0:5β2m þ 0:25β2n ; ϑ7 ¼  1:5β2m ; ϑ8 ¼  0:5βm ; ϑ9 ¼ 0:25βm β2n þ 0:25β3m  0:25βm ; ϑ10 ¼ βm ; ϑ11 ¼ 0:25½β2m ð2 β2n Þ þ β2n ; ϑ12 ¼  0:25βm ðβ2n þ β2m  1Þ; ϑ13 ¼ 0:25βm ðβ2n  β2m  1:5Þ; ϑ14 ¼  βm ; ϑ15 ¼ β2m ðβ2n þ 0:25Þ; ϑ16 ¼ β2m ;

η12 ¼ 27ðb21 b12  4b22 Þβ3m þ 6ðb11 þ b21  b12 b22 Þβm þ3ð4b31 þ3b12 þ b21 Þβm β2n ; η13 ¼ 256b22 β4m 16ð3b12  3b21  b11 þ5b22 Þβ2m ;

ϑ1 η1  ϑ2 η2 ϑ1 η2 þ ϑ2 η1 ϑ3 η3 ϑ4 η4 ; A2 ¼ ; A3 ¼ ; A1 ¼ η21 þ η22 η21 þ η22 η23 þη24 ϑ3 η4 þ ϑ4 η3 ϑ5 η 5  ϑ8 η 6 ϑ6 η5  ϑ8 η6 A4 ¼ ; A51 ¼ ; A52 ¼ ; η23 þ η24 η25 þ η26 η25 þ η26 ϑ7 η5  ϑ10 η6 ϑ5 η6 þ ϑ8 η5 ϑ6 η6 þ ϑ9 η5 A53 ¼ ; A61 ¼ ; A62 ¼ ; η25 þ η26 η25 þ η26 η25 þ η26 ϑ7 η6 þ ϑ10 η5 ϑ11 η7  ϑ12 η8 ϑ11 η8 þ ϑ12 η7 A63 ¼ ; A7 ¼ ; A8 ¼ ; η25 þ η26 η27 þ η28 η27 þ η28 ϑ13 η9  ϑ15 η10 ϑ14 η9  ϑ16 η10 ϑ13 η10 þ ϑ15 η9 A91 ¼ ; A92 ¼ ; A101 ¼ ; η29 þ η210 η29 þ η210 η29 þ η210

ϑ19 η13  ϑ20 η14 ϑ19 η14 þ ϑ20 η13 ; A14 ¼ ; η213 þ η214 η213 þ η214 ϑ21 ϑ22 ; A16 ¼ : A15 ¼ η15 η16

þ9ð2b31 þb21 þ b12 Þβ2m β2n  2ðb31 þ b21 þ b11 Þβ2n þ b11 β4n ;

ð21Þ

where the following definitions apply:

A11 ¼

ϑ20 ¼ βm ð0:25  β2m Þ;

ϑ21 ¼ 0:25β2n ðβ2m þ1Þ;

2

 sin ð2βm zÞ þ A7 f 1 cos ð2βm zÞ cos ð2βn θ1 Þ

ϑ14 η10 þ ϑ16 η9 ; η29 þ η210

ϑ18 ¼ β2m ðβ2n 3:25Þ;

þ8ð4b31 þ3b12 þ b21 Þβm β2n ;

þ ðA51 f 2 þA52 f 1 þ A53 f 2 Þ cos ð2βm zÞ þ ðA61 f 2 þ A62 f 1 þ A63 f 2 Þ

A102 ¼

181

η14 ¼ 64ðb21 b12  4b22 Þβ3m þ 8ðb11 þ b21  b12 b22 Þβm ; η15 ¼ 8ðb31 þ b21 þb11 Þβ2n þ 16b11 β4n ; η16 ¼ 11b22  6b12 þ 6b21  3b11

ð23Þ

Lets apply the Galerkin method to Eq. (17): Z z0 Z 2π sin γ ½L11 ðΨ 1 Þ þ L12 ðwÞ 0

0

þ L13 ðΨ 1 ; wÞez sin ðβm zÞ sin ðβn θ1 Þ dθ1 dz ¼ 0 Z 0

z0

Z

2π sin γ 0

ð24Þ

½L11 ðΨ 1 Þ þ L12 ðwÞ þ L13 ðw; wÞez sin2 ðβm zÞ dθ1 dz ¼ 0 ð25Þ

Introducing (19) and (21) into Eqs. (24) and (25), after the integration and some rearrangements yields the following equations: 2

3

Tf 1 þ Δ1 f 1 þ K wp1 f 1 þ Δ4 f 2 f 1 þ Δ5 f 1 þ ðΔ2 þ Δ3 þ Δ6 Þf 2 f 1 ¼ 0 2

3

ð26Þ

2

T½Δ7 f 2 þΔ8  þ K wp2 f 2 þ Δ9 f 2 þ ðΔ11 þ Δ13 Þf 1 þ Δ12 f 2 þ Δ14 f 1 f 2 2

þ ðΔ15 þΔ10 Þf 2 ¼ 0

ð27Þ

where f 1 ¼ f 1 =h and f 2 ¼ f 2 =h are dimensionless amplitudes and the following definitions apply: Δ1 ¼ Δ1 ;

Δ2 ¼ Δ2 h;

Δ6 ¼ Δ6 h;

Δ7 ¼ hΔ7 ; 2

Δ11 ¼ Δ11 h ;

Δ3 ¼ Δ3 h;

2

Δ4 ¼ Δ4 h ;

Δ8 ¼ Δ8 ; Δ9 ¼ Δ9 h; 3

Δ12 ¼ Δ12 h ;

2

Δ13 ¼ Δ13 h ;

2

Δ5 ¼ Δ5 h ; 2

Δ10 ¼ Δ10 h ; 3

Δ14 ¼ Δ14 h ;

2

Δ15 ¼ Δ15 h ; K wp2 ¼ K wp2 h;

K wp2 ¼ ðQ 3 K w þ Q 4 K p Þ=B02 ;

K wp1 ¼ ðQ 1 K w þ Q 2 K p Þ=B01

ð28Þ

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A.H. Sofiyev / Thin-Walled Structures 80 (2014) 178–191

shells without an elastic foundation:

in which Δ1 ¼ Δ001 =B01 ;

Δ2 ¼ Δ002 =B01 ;

Δ3 ¼ Δ003 =B01 ;

Δ4 ¼ Δ004 =B01 ;

Δ5 ¼ Δ005 =B01 ; Δ6 ¼ Δ006 =B01 ; Δ7 ¼ Δ007 =B02 ; Δ8 ¼ Δ008 =B02 ; Δ9 ¼ Δ009 =B02 ; Δ10 ¼ Δ0010 =B02 ; Δ11 ¼ Δ0011 =B02 ; Δ12 ¼ Δ0012 =B02 ; Δ13 ¼ Δ0013 =B02 ; Δ14 ¼ Δ0014 =B02 ; Δ15 ¼ Δ0015 =B02 ; S2 β2 ðβ2 þ 2Þðe4z0  1Þ S21 ðe4z0  1Þ ; B02 ¼ ; B01 ¼ 1 m m 2 8ðβm þ4Þ 16ð4 þ β2m Þð1 þ β2m Þ S4 β2 ðe6z0  1Þ S2 β2 ðβ2 þ β2n þ 3Þðe4z0 1Þ ; Q2 ¼  1 m m ; Q1 ¼  1 m 2 12ðβm þ 9Þ 8ðβ2m þ4Þ S41 β4m ðe6z0  1Þ S2 β4 ð4β2 þ 1Þðe4z0  1Þ ; Q4 ¼  1 m 2 m : Q3 ¼  2 2 2ðβm þ 9Þð4βm þ 9Þ 16ðβm þ 4Þðβ2m þ 1Þ

λ¼

ð29Þ

3 2

þ ½Δ1 Δ7  ðΔ2 þ Δ3 þ Δ6 ÞΔ8 þ Δ9 þ K wp2 þ ð  Δ5 Δ17 þ Δ14 Þf 1 f 2 ¼0

ð30Þ

From Eq. (30), the following relationship between the symmetrical and unsymmetrical components of deflection, w, is obtained: 2

f 2 ¼ λwp f 1

ð31Þ

where λwp ¼

Δ11 þ Δ13  Δ8 Δ5 ðΔ2 þ Δ3 þ Δ6 ÞΔ8 þ ðΔ1 þK wp1 ÞΔ7  Δ9 K wp2

ð32Þ

Introduction of (30) into Eq. (26), we obtain the following equation for the heterogeneous orthotropic truncated conical shell on a Pasternak elastic foundation: 2

4

2 T low 1wp ¼  Δ1a  K wp1  ½Δ5a þ ðΔ2a þ Δ3a þ Δ6a Þλwp f 1  Δ4a λwp f 1

ð33Þ

where the following notations are introduced: low T low 1wp ¼ T wp =E 0S h;

K wp1 ¼ K wp1 =E0S h;

Δia ¼ Δi =E0S h; i ¼ 1; 2; …; 6 ð34Þ 2

Minimizing Eq. (33) depending on f 1 , we obtain the following equation for the dimensionless lower (or nonlinear) critical axial load of heterogeneous orthotropic truncated conical shells on a Pasternak elastic foundation: T low 1crwp ¼  Δ1a K wp1 þ

½Δ5a þ ðΔ2a þ Δ3a þ Δ6a Þλwp 2 4Δ4a λ2wp

ð35Þ

As K p ¼ 0 is taken into account in Eq. (35), the following equation is obtained for the dimensionless lower (or nonlinear) critical axial load of heterogeneous orthotropic truncated conical shells surrounded by the Winkler elastic foundation, T low 1crw ¼ Δ1a  K w1 þ

½Δ5a þ ðΔ2a þ Δ3a þ Δ6a Þλw 2 4Δ4a λ2w

Q 1K w ; B01 Δ11 þ Δ13  Δ8 Δ5 λw ¼ ; ðΔ2 þ Δ3 þ Δ6 ÞΔ8 þ ðΔ1 þ K w1 ÞΔ7  Δ9 K w2 Q Kw K w2 ¼ hK w2 ; K w2 ¼ 3 B02

Δ11 þ Δ13  Δ8 Δ5 ðΔ2 þ Δ3 þ Δ6 ÞΔ8 þ Δ1 Δ7  Δ9

ð39Þ

T up 1crwp ¼  Δ1a  K wp1

ð40Þ

where up T up 1crwp ¼ T crwp =E 0S h

ð41Þ

When K p ¼ 0 is considered in Eq. (40), we obtain the following equation for the dimensionless upper (or linear) critical axial load of truncated conical shells on a Winkler elastic foundation: T up 1crw ¼  Δ1a  K w1

ð42Þ

where up T up 1crw ¼ T crw =E 0S h

ð43Þ

As K w ¼ K p ¼ 0, from Eq. (40), we obtain the following equation for the dimensionless upper critical axial load of truncated conical shells without an elastic foundation: T up 1cr ¼ 

Δ01 B01 E0S h

ð44Þ

where up T up 1cr ¼ T cr =E 0S h

ð45Þ

If φðζÞ ¼ 1 is written in Eqs. (35), (36), (38) and (40), (42), (44), the convenient expressions for homogenous orthotropic truncated conical shells in non-liner and linear cases, respectively, will be obtained in a special case. As φðζÞ ¼ 1; E0S ¼ E0θ ¼ E0 ; νSθ ¼ νθS ¼ ν0 are taken into account in Eqs. (35), (36), (38) and (40), (42), (44), the appropriate expressions for homogenous isotropic truncated conical shells in the non-liner and linear cases, respectively, will be found in a special case. Eqs. (35), (36), (38) and (40), (42), (44) will be minimized with respect to the wave numbers ðm; nÞ, respectively, for getting the minimum values of lower and upper critical axial compressive loads of the heterogeneous orthotropic truncated conical shells with and without elastic foundation.

6. Numerical analysis 6.1. Comparative studies

ð36Þ

where K w1 ¼ K w1 =E0S h;

ð38Þ

After elimination of the nonlinear terms in Eq. (35), we obtain the following expression for the dimensionless upper (or linear) critical axial compressive load for the heterogeneous orthotropic truncated conical shell on a Pasternak foundation:

ð  Δ1 Δ7 þ Δ12 Þf 2 þ ½  ðΔ2 þ Δ3 þ Δ6 ÞΔ7  Δ4 Δ8 þ Δ15 þΔ10 f 2 2 þ ð  Δ5 Δ8 þ Δ11 þ Δ13 Þf 1 þ Δ1 Δ8

½Δ5a þ ðΔ2a þ Δ3a þΔ6a Þλ2 4Δ4a λ2

where

and all symbols used in Eqs. (26)–(29) are described in detail in Appendix B. The expression T which was given in Eq. (26), is substituted into Eq. (27) results that 3

T low 1cr ¼  Δ1a þ

K w1 ¼

ð37Þ

When K w ¼ K p ¼ 0 is assumed in Eq. (35), the following equation is obtained for the dimensionless lower (or nonlinear) critical axial load of heterogeneous orthotropic truncated conical

To validate the results of the present study, some comparison studies are presented. Table 1 shows the comparison of the values up of upper critical axial load ratios, T up cr =T cl , of homogeneous orthotropic and isotropic cylindrical shells without an elastic foundation, which are obtained in this study using Eq. (45), with those obtained by Prado et al. [6] using Eq. (22), in which internal flowing fluid effect is neglected (see, Eq. (22) in Ref. [5]). As γ ¼ π=18; 000-0 yields z0 ¼ lnð1 þ L=S1 Þ-0; βm sin γ ¼ mπR=L1 ; R1 ¼ R2 ¼ R; L ¼ L1 , i.e., an orthotropic truncated conical shell coincides with an orthotropic cylindrical shell. Here L1 and R are length and radius of the orthotropic cylindrical shell, respectively. The related values of a classical critical axial load of the orthotropic

A.H. Sofiyev / Thin-Walled Structures 80 (2014) 178–191

183

Table 1 up Comparison of the ratios, T up cr =T cl , for different material properties of the cylindrical shell. Case

1 2 3 4 5

Material properties νSθ

νθS

E0S  1010 ðN=m2 Þ

E0θ  1010 ðN=m2 Þ

G0  1010 ðN=m2 Þ

0.131926 0.131926 0.131926 0.04 0.012114

0.012114 0.04 0.131926 0.131926 0.131926

22.7350 6.8599 2.0545 2.0799 2.0876

2.0876 2.0799 2.0545 6.8599 22.7350

0.7958 0.7958 0.7958 0.7958 0.7958

up T up cr =T cl ððmcr ; ncr ÞÞ

Case

Prado et al. [6]

Present study

1 2 3 4 5

0.271(1,7) 0.639(2,9) 1.624(3,9) 2.121(4,9) 2.816(5,8)

0.271(1,7) 0.639(2,9) 1.624(3,9) 2.121(4,9) 2.816(5,8)

Table 2 Comparison of T low 1cr for isotropic truncated conical shell without an elastic foundation. γ (deg.)

10 15 20

Agamirov [3]

Present study

3 T low 1cr ðncr ; ξ; f T Þ  10

3 T low 1cr ðmcr ; ncr Þ  10

1.060(11,1.1,2.5) 1.170(10,1.1,1.8) 1.250(10,1.1,1.0)

1.086(18,18) 1.182(19,5) 1.255(26,14)

2

cylindrical shell are obtained from the expression T up ¼ E0S h = cl pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 3ð1  νSθ 2 Þ. The orthotropic and isotropic material properties are taken from the study of Prado et al. [6] and presented in Table 1. The current cylindrical shell characteristics are taken to be R=h ¼ 100 and L1 =R ¼ 1. It is seen that the results of the present study are in excellent agreement with the results of Prado et al. [6]. As a second example, the values of T low 1cr for the homogeneous isotropic truncated conical shell without an elastic foundation are compared with the results of Ref. [3], and presented in Table 2. Computations have been carried out for the following data: E0 ¼ 2:1  105 MPa; ν0 ¼ 0:3; h ¼ 5  10  4 m; R2 ¼ 0:15 m; h1 =R2 ¼ 2; ξ ¼ βm =βn . Here h1 and f T are the height of truncated conical shell and total amplitude. The values of the lower critical axial load of Ref. [3], are slightly lower than the current results due to the consideration of the initial imperfection ðf 0 ¼ 0:001Þ: 6.2. Nonlinear buckling analysis for heterogeneous orthotropic truncated conical shells with and without an elastic foundation After verifying the accuracy and reliability of this method, we will apply it for nonlinear buckling analysis of heterogeneous orthotropic conical shells with and without Pasternak and Winkler elastic foundations. Numerical calculations were performed using the expressions (35), (36), (38) and (40), (42), (44). In computational analysis, we consider two cases for heterogeneity functions: (a) The exponential function, i.e., φðζÞ ¼ eηðζ  0:5Þ [10,15], where η is a heterogeneity parameter and positive real number. q (b) The power-law function, i.e., φðζÞ ¼ 1 þ μζ [19,44], where μ is heterogeneity coefficient, satisfying 0 r μ r 1 and q ¼ 1; 2 is the power-law index.

In all subsequent tables and figures, heterogeneity functions are taken as linear, quadratic and exponential functions and it is assumed that μ ¼ 1 and η ¼ 1. Maple 14 software program is used for the calculations and the results are presented in Tables 3 and 4 and Figs. 2 and 3. Here, “up” and “low” indicate upper (or linear) and lower (or nonlinear), “H” and “HT” indicate homogeneous and heterogeneous, and “wp” and “w” indicate Pasternak and Winkler elastic foundations, respectively. 6.2.1. The effect of degrees of orthotropy Fig. 2 illustrates the effect of degrees of orthotropy ðE0S =E0θ Þ on the values of dimensionless upper and lower axial buckling loads of homogeneous and heterogeneous orthotropic truncated conical shells with and without a Pasternak foundation. The following orthotropic material properties, Pasternak foundation stiffness and conical shell characteristics are considered: E0S ¼ 2  1011 (Pa), E0θ ¼ E0S =i; i ¼ 10; 25; 40; νSθ ¼ 0:2 [4], ðK w ; K p Þ ¼ ð1  106 N=m3 ; 1  104 N=mÞ and h¼0.01 m; L=R1 ¼ 2; R1 =h ¼ 150; γ ¼ 45 3 , respectively. As the ratio, E0S =E0θ , increases, the values of dimensionless upper and lower critical axial loads of H and HT orthotropic conical shells with and without a Pasternak elastic foundation decrease. The influence of a Pasternak foundation on the values of upper dimensionless critical axial loads for H and HT orthotropic truncated conical shells increases monotonically, whereas, this effect on the dimensionless lower critical axial loads vary irregularly, as E0S =E0θ increases. The influence of heterogeneity on the values of upper dimensionless critical axial loads for orthotropic truncated conical shells with and without a Pasternak foundation decreases usually, as E0S =E0θ increases. The influence of heterogeneity on the values of lower dimensionless critical axial loads for orthotropic truncated conical shells surrounded by a Pasternak foundation is significantly higher than the dimensionless upper critical axial loads usually, but changes irregularly, as E0S =E0θ increases. The change interval of up T low 1cr =T 1cr (or effect of geometric nonlinearity) for homogeneous orthotropic conical shell surrounded by a Pasternak foundation is in the range (0.48, 0.56), while for heterogeneous orthotropic conical shells these intervals are (0.39, 0.56), (0.50, 0.63) and (0.43, 0.52) with the linear, quadratic and exponential profiles, respectively, as E0S =E0θ increases from 10 to 40. In addition, for unconstrained homogeneous orthotropic conical shell this interval is (0.44, 0.47) and for heterogeneous orthotropic conical shells these intervals are (0.39, 0.47), (0.48, 0.52) and (0.46, 0.49) with the linear, quadratic and exponential profiles, respectively, as E0S =E0θ increases from 10 to 40. When the variation of Young’s moduli is given by linear and quadratic functions, it was found that the effect of heterogeneity on the dimensionless critical axial loads is greatest for being quadratic

184

A.H. Sofiyev / Thin-Walled Structures 80 (2014) 178–191

Table 3 Variation of dimensionless critical axial loads for HT and H orthotropic truncated conical shells with and without a Pasternak foundation versus K w and K p . Kw (N/m3)

Kp (N/m)

H

HT linear

3 T up 1cr  10 (m,n)

3 T low 1cr  10 (m,n)

3 T up 1cr  10 (m,n)

3 T low 1cr  10 (m,n)

0

0.693(4,13)

0.348(9,37)

0.662(4,13)

0.286(6,10)

0 7.5  103 104 5  104

0.782(5,12) 0.808(6,12) 0.815(6,15) 0.930(6,11)

0.392(5,6) 0.414(4,14) 0.432(8,25) 0.460(7,19)

0.749(5,13) 0.773(6,12) 0.780(6,12) 0.897(6,11)

0.332(7,38) 0.374(4,14) 0.370(4,14) 0.421(6,19)

106

0 7.5  103 104 5  104

0.803(6,12) 0.826(6,12) 0.833(6,12) 0.947(7,9)

0.369(5,6) 0.372(8,35) 0.395(10,33) 0.431(5,5)

0.768(6,12) 0.791(6,12) 0.798(6,12) 0.911(7,10)

0.299(7,20) 0.344(9,37) 0.390(4,14) 0.413(6,19)

2.5  106

0 7.5  103 104 5  104

0.904(7,11) 0.923(7,10) 0.929(7,10) 1.027(7,9)

0.409(6,10) 0.447(10,33) 0.454(9,29) 0.539(6,19)

0.864(7,11) 0.884(7,11) 0.890(7,11) 0.990(7,10)

0.388(8,36) 0.417(7,37) 0.433(6,14) 0.465(10,32)

Kw (N/m3)

Kp (N/m)

HT quadratic

0 7.5  10

5

HT exponential

3 T up 1cr  10 (m,n)

3 T low 1cr  10 (m,n)

3 T up 1cr  10 (m,n)

3 T low 1cr  10 (m,n)

0

0

0.772(4,12)

0.375(1,9)

0.524(4,13)

0.207(3,12)

7.5  105

0 7.5  103 104 5  104

0.864(5,12) 0.891(5,12) 0.900(5,12) 1.017(6,10)

0.401(8,35) 0.426(7,35) 0.436(7,35) 0.462(5,17)

0.556(8,1) 0.570(8,1) 0.574(8,1) 0.649(8,1)

0.218(6,41) 0.250(6,24) 0.283(6,40) 0.357(6,27)

106

0 7.5  103 104 5  104

0.890(5,12) 0.914(6,11) 0.921(6,11) 1.034(6,10)

0.399(8,35) 0.430(7,35) 0.438(7,35) 0.464(5,17)

0.566(8,1) 0.580(8,1) 0.585(8,1) 0.659(8,1)

0.225(6,41) 0.252(6,24) 0.291(6,40) 0.362(6,27)

2.5  106

0 7.5  103 104 5  104

0.999(7,9) 1.017(7,10) 1.023(7,10) 1.120(7,9)

0.536(5,6) 0.547(10,39) 0.606(4,14) 0.607(6,15)

0.626(9,1) 0.640(9,1) 0.644(9,1) 0.719(9,1)

0.339(6,24) 0.374(6,40) 0.385(15,50) 0.391(6,27)

and least for being linear. When linear, quadratic and exponential profiles are compared with each other the highest effect of heterogeneity on the dimensionless critical axial loads is observed in the exponential profile. 6.2.2. Different foundation models The variation of the values of dimensionless upper and lower critical axial loads for H and HT orthotropic conical shells surrounded by Winkler and Pasternak elastic foundations with different K w and K p are presented in Table 3. In numerical computations, material properties and conical shell parameters are taken to be E0S = E0θ ¼ 3; G0 = E0θ ¼ 0:5; νSθ ¼ 0:25 and h ¼ 0:01 m; L=R1 ¼ 2; R1 =h ¼ 150; γ ¼ 45 3 , respectively. In Table 3, the values of dimensionless upper and lower critical axial loads are multiplied by 103 . The results show that the buckling strength of orthotropic truncated conical shells surrounded by a Winkler elastic foundation is lower than that for the conical shells surrounded by a Pasternak elastic foundation. It can be seen from Table 3, with increasing the values of K w and K p , the dimensionless critical loads and corresponding circumferential wave numbers increase. Influences of Pasternak or Winkler foundations on the dimensionless critical loads of H and HT orthotropic truncated conical shells increase with increasing K w and K p together or separately. The influence of heterogeneity on the dimensionless critical axial loads of orthotropic truncated conical shells decreases with increasing K w and K p together or separately. 6.2.3. The effect of variation of the semi-vertex angle of the conical shell The variations of the values of dimensionless upper and lower critical axial loads for H and HT (a) graphite/epoxy and

(b) glass/epoxy truncated conical shells with and without a Pasternak foundation depending on the semi-vertex angle, γ, are presented in Table 4. The following glass/epoxy and graphite/ epoxy orthotropic material properties are taken E0S ¼ 5:37791  1010 Pa; E0θ ¼ 1:79264  1010 Pa; G0 ¼ 8:9632  109 Pa; νSθ ¼ 0:25 and E0S ¼ 1:724  1011 Pa; E0θ ¼ 7:79  109 Pa; νSθ ¼ 0:35, respectively [47]. The truncated shell characteristics are h ¼ 0:01 m; L=R1 ¼ 2; R1 =h ¼ 150. The stiffness of a Pasternak elastic foundation is taken (K w ; K p )¼ (1  106 N/m3; 1  104 N/m). It is noted that (K w ; K p )¼ (0; 0) is considered for the unconstrained shell. The semi-vertex angle with four values (γ ¼ 151; 301; 451; 601) is taken into account. It is observed that with increase of the semi-vertex angle γ, the upper and lower values of the dimensionless critical axial loads for H and HT graphite/epoxy and glass/epoxy conical shells with and without a Pasternak elastic foundation decrease, while the change in the wave numbers are irregular. The influence of a Pasternak foundation on the values of upper and lower dimensionless critical axial loads for H and HT graphite/epoxy or glass/epoxy truncated conical shells increases, as the semi-vertex angle, γ, increases. In addition, the effect of Pasternak foundation on the values of the upper dimensionless critical axial loads for glass/epoxy truncated conical shells is influential in the small values of γ, whereas, this effect is influential in the high values of γ for graphite/epoxy truncated conical shells. The influence of heterogeneity on the values of upper dimensionless critical axial loads for graphite/epoxy truncated conical shells surrounded by a Pasternak foundation decreases usually for all heterogeneity functions, whereas, this effect increases for glass/epoxy truncated conical shells except for the quadratic profile, as the semi-vertex angle, γ, increases. The influence of heterogeneity on the values of lower dimensionless critical axial loads for graphite/epoxy or

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185

Table 4 Variation of dimensionless upper and lower axial buckling loads and corresponding wave numbers for H and HT conical shells with and without an elastic foundation versus γ. γ(angle) (deg.)

H

HT linear

3 T up 1cr  10 (m,n)

Graphite/epoxy, ðK w ; K p Þ ¼ ð1  106 N=m3 ; 1  104 N=mÞ 15 0.637(6,1) 30 0.479(5,5) 45 0.347(5,1) 60 0.250(4,1) γ(angle) (deg.)

3 T up 1cr  10 (m,n)

3 T low 1cr  10 (m,n)

0.396(6,13) 0.248(6,17) 0.164(4,7) 0.144(7,33)

0.611(6,2) 0.458(6,1) 0.332(5,1) 0.241(4,1)

0.301(6,12) 0.168(6,16) 0.160(6,21) 0.116(4,11)

HT quadratic T up 1cr

3

 10 (m,n)

Graphite/epoxy, ðK w ; K p Þ ¼ ð1  106 N=m3 ; 1  104 N=mÞ 15 0.708(6,1) 30 0.530(5,5) 45 0.384(5,1) 60 0.273(4,1) γ(angle) (deg.)

3 T low 1cr  10 (m,n)

HT exponential T low 1cr

3

 10 (m,n)

0.473(3,8) 0.304(5,11) 0.206(5,15) 0.194(2,2)

H

3 T up 1cr  10 (m,n)

3 T low 1cr  10 (m,n)

0.401(6,1) 0.305(6,1) 0.228(5,1) 0.175(5,1)

0.196(8,23) 0.169(5,10) 0.115(6,21) 0.103(4,11)

HT linear

3 T up 1cr  10 (m,n)

3 T low 1cr  10 (m,n)

3 T up 1cr  10 (m,n)

3 T low 1cr  10 (m,n)

Graphite/epoxy, ðK w ; K p Þ ¼ ð0; 0Þ 15 30 45 60

0.614(4,12) 0.443(3,13) 0.300(1,10) 0.146(1,5)

0.349(6,13) 0.187(6,17) 0.140(7,23) 0.082(3,5)

0.588(5,10) 0.424(4,12) 0.288(1,10) 0.144(1,5)

0.256(8,23) 0.108(6,16) 0.094(5,14) 0.079(4,12)

γ(angle) (deg.)

HT quadratic

HT exponential

3 T up 1cr  10 (m,n)

3 T low 1cr  10 (m,n)

3 T up 1cr  10 (m,n)

3 T low 1cr  10 (m,n)

Graphite/epoxy, ðK w ; K p Þ ¼ ð0; 0Þ 15 30 45 60

0.685(5,9) 0.494(4,11) 0.334(1,10) 0.159(1,5)

0.410(3,8) 0.236(9,35) 0.159(4,8) 0.099(5,20)

0.379(4,12) 0.273(3,13) 0.185(1,10) 0.092(1,5)

0.181(7,18) 0.138(6,17) 0.072(7,28) 0.044(4,12)

γ(angle) (deg.)

H 3 T up 1cr  10 (m,n)

Glass/epoxy, ðK w ; K p Þ ¼ ð1  106 N=m3 ; 1  104 N=mÞ 15 1.477(6,12) 30 1.119(6,13) 45 0.833(6,12) 60 0.598(6,9) γ(angle) (deg.)

3 T low 1cr  10 (m,n)

3 T up 1cr  10 (m,n)

3 T low 1cr  10 (m,n)

0.761(7,3) 0.654(10,29) 0.395(10,33) 0.311(7,31)

1.419(6,12) 1.072(6,13) 0.798(6,12) 0.573(6,9)

0.740(10,24) 0.509(8,38) 0.390(4,14) 0.290(4,40)

HT quadratic 3 T up 1cr  10 (m,n)

Glass/epoxy, ðK w ; K p Þ ¼ ð1  106 N=m3 ; 1  104 N=mÞ 15 1.640(6,12) 30 1.240(6,12) 45 0.921(6,11) 60 0.659(6,8) γ(angle) (deg.)

HT linear

HT exponential 3 T low 1cr  10 (m,n)

3 T up 1cr  10 (m,n)

3 T low 1cr  10 (m,n)

0.900(9,34) 0.709(11,39) 0.438(7,35) 0.321(4,38)

1.077(10,1) 0.802(9,1) 0.585(8,1) 0.412(7,1)

0.501(12,30) 0.417(11,32) 0.291(6,40) 0.201(5,22)

Homogeneous

HT linear

3 T up 1cr  10 (m,n)

3 T low 1cr  10 (m,n)

3 T up 1cr  10 (m,n)

3 T low 1cr  10 (m,n)

Glass/epoxy, ðK w ; K p Þ ¼ ð0; 0Þ 15 30 45 60

1.372(5,12) 0.996(4,13) 0.693(4,13) 0.433(3,11)

0.719(18,48) 0.498(1,6) 0.348(9,37) 0.217(7,32)

1.315(5,13) 0.953(4,13) 0.662(4,13) 0.415(3,11)

0.629(8,39) 0.476(12,36) 0.286(6,10) 0.191(7,24)

γ(angle) (deg.)

HT quadratic

Glass/epoxy, ðK w ; K p Þ ¼ ð0; 0Þ 15 30 45 60

HT exponential

3 T up 1cr  10 (m,n)

3 T low 1cr  10 (m,n)

3 T up 1cr  10 (m,n)

3 T low 1cr  10 (m,n)

1.530(5,12) 1.113(4,13) 0.772(4,12) 0.483(3,11)

0.739(9,21) 0.535(6,9) 0.375(1,9) 0.229(4,6)

1.040(8,10) 0.755(7,11) 0.524(4,13) 0.329(4,11)

0.439(8,38) 0.361(1,7) 0.207(3,12) 0.140(5,21)

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A.H. Sofiyev / Thin-Walled Structures 80 (2014) 178–191

0.4

0.3 H HT Lin HT Quad HT Exp

0.3

graphite/epoxy

H wp HT Lin wp HT Quad wp HT Exp wp

H HT Lin HT Quad HT Exp

H wp HT Lin wp HT Quad wp HT Exp wp

0.2 0.2 0.1 0.1

0 10

25

40

0 100

150

E0S/E0θ

0.7

0.6 H HT Lin HT Quad HT Exp

200

250

300

R1/h

H wp HT Lin wp HT Quad wp HT Exp wp

glass/epoxy

0.6

H HT Lin HT Quad HT Exp

0.5

H wp HT Lin wp HT Quad wp HT Exp wp

0.4 0.4 0.3 0.2

0.2 0.1

0 10

25

40

E0S/E0θ Fig. 2. Variation of upper and lower dimensionless critical axial loads for H and HT orthotropic conical shells with and without a Pasternak elastic foundation versus E0S =E0θ :

glass/epoxy truncated conical shells surrounded by a Pasternak foundation is significantly higher than the dimensionless upper critical axial loads usually, but changes irregularly, as the semivertex angle, γ, increases. It can also be seen that the influences of Pasternak foundation and heterogeneity on the values of lower dimensionless critical axial loads are sometimes influential in glass/epoxy or graphite/epoxy shells. The longitudinal wave numbers corresponding to critical axial loads for orthotropic truncated conical shells surrounded by a Pasternak foundation is higher than the number of longitudinal waves for unconstrained orthotropic truncated conical shells, while the circular wave numbers is small, as 01 o γ r 601. It is found that the ratio of lower and upper dimensionless critical axial loads with and without on an elastic up foundation ðT low 1cr =T 1cr Þ varies irregularly, as the semi-vertex angle, γ, increases. It is noted that determination of the effects of heterogeneity and Pasternak elastic foundation are very difficult in the large deformation and each case should be considered separately. The values of the dimensionless critical axial load of graphite/epoxy conical shell are higher than those for the glass/ up epoxy conical shell. The ratio, T low 1cr =T 1cr , is more regular for glass/ epoxy shells than graphite/epoxy shells.

6.2.4. The effect of variation of the radius-to-thickness ratio Figs. 3 and 4 depict the influences of the Pasternak foundation and heterogeneity on the upper and lower dimensionless axial

0 100

150

200

250

300

R1/h Fig. 3. Variation of dimensionless lower critical axial loads for HT and H, graphite/ epoxy and glass/epoxy conical shells with and without a Pasternak foundation versus R1 =h:

buckling loads for (a) graphite/epoxy and (b) glass/epoxy truncated conical shells versus the ratio,R1 =h. To construct Figs. 3 and 4, we have used the same material properties and the foundation stiffness, which were used in Table 4. In addition, the conical shell characteristics are taken to be h ¼ 0:01 m; L=R1 ¼ 2; γ ¼ 45 3 . The ratio, R1 =h, increases from 100 to 300 by step 50. It is observed that the upper and lower dimensionless axial buckling loads for graphite/epoxy and glass/epoxy truncated conical shells with and without a Pasternak foundation decrease, while and corresponding longitudinal wave numbers increase and corresponding circumferential wave numbers vary irregularly, as R1 =h increases. The influence of heterogeneity on the values of upper dimensionless critical axial loads for graphite/epoxy and glass/epoxy truncated conical shells surrounded by elastic foundation decreases usually, while this influence on the values of dimensionless lower critical axial loads changes irregularly, when R1 =h increases. For example, as the values of the upper and lower dimensionless axial buckling loads for HT (linear, quadratic and exponential profiles) graphite/ epoxy conical shells surrounded by a Pasternak foundation compared with those for homogenous graphite/epoxy conical shells, the influences of heterogeneity on the values of upper and lower dimensionless axial buckling loads are observed as follows: 2.12%, ( 4.24%), 34.75% and 5.98%, ( 18.73%), 39.84% for linear, quadratic, exponential profiles, respectively, when R1 =h ¼ 100, while 4.05%, ( 9.46%), 28.83% and 10.19%, (  17.59%), 33.33% for linear, quadratic and exponential profiles, respectively, when R1 =h ¼ 300. For glass/epoxy conical shells, the influences of heterogeneity on

A.H. Sofiyev / Thin-Walled Structures 80 (2014) 178–191

the values of upper and lower dimensionless axial buckling loads are 3.90%, ( 10.74%), 27.71% and 4.14%, (  11.15%), 13.13% for linear, quadratic, exponential profiles, respectively, when R1 =h ¼ 100, while 4.08%, ( 9.71%), 30.68% and 5.14%, (  9.09%), 31.23% for linear, quadratic, exponential profiles, respectively, when R1 =h ¼ 300. The variation of lower to upper critical axial load ratios (effect of nonlinearity) for H and HT glass/epoxy truncated conical shells with and without a Pasternak foundation is more irregular than the graphite/epoxy truncated conical shells, as R1 =h increases. The comparison of critical loads of graphite/ epoxy conical shell with the glass/epoxy conical shell showed that the values of upper and lower critical axial loads for the graphite/ epoxy conical shell are lower than those for the glass/epoxy conical shell.

graphite/epoxy

0.5

H HT Lin HT Quad HT Exp

0.4

187

H wp HT Lin wp HT Quad wp HT Exp wp

0.3

0.2

0.1

0 100

150

200

250

300

7. Conclusions

R1/h

The buckling problem of a heterogeneous orthotropic truncated conical shell subjected to an axial load and surrounded by elastic media is analyzed based on the finite deformation theory. Using the von-Karman nonlinearity, the basic equations of elastic stability of the heterogeneous orthotropic truncated conical shell surrounded by an elastic medium are derived. The basic equations are solved using superposition and Galerkin methods and obtained expressions for upper and lower critical axial loads heterogeneous orthotropic truncated conical shell with and without on an elastic foundation. The results of numerical calculations are presented in tables and figures, which show the influences of elastic foundations, heterogeneity, orthotropy and geometric parameters on the values of the upper and lower critical loads of the truncated conical shell.

glass/epoxy

1.3

H HT Lin HT Quad HT Exp

1.1

H wp HT Lin wp HT Quad wp HT Exp wp

0.9

0.7

0.5

0.3

0.1 100

150

200

250

300

R1/h Fig. 4. Variation of dimensionless upper critical axial loads for HT and H, graphite/ epoxy and glass/epoxy conical shells with and without a Pasternak foundation versus R1 =h:

Acknowledgment The author thanks Scientific and Technical Research Council of Turkey for the support of the project 110M695.

Appendix A In Eqs. (17) and (18), Lij ði ¼ 1; 2; 3Þ are differential operators and are given as ∂4 Ψ 1 ∂3 Ψ 1 ∂2 Ψ 1 ∂Ψ 1 L11 ðΨ 1 Þ ¼ c12 e2z 4 þ ðc11  c22 þ 4c12 Þe2z 3 þ ð5c12 þ 3c11 3c22  c21 Þe2z 2 þ 2ðc11  c22 þ c12  c21 Þe2z ∂z ∂z ∂z ∂z
2  4 4 3 ∂ Ψ 1 ∂Ψ 1 ∂ Ψ ∂ Ψ ∂ Ψ ∂2 Ψ 1 1 1 1 þ 2Ψ 1 S1 e3z cot γ þc21 e2z 4 þ ðc11  2c31 þ c22 Þe2z þ þ ðc11  4c31 þ3c22 Þe2z þ 2ðc22  c31 þ c21 Þe2z 2 þ ∂z ∂z2 ∂θ1 ∂z2 ∂θ21 ∂z∂θ21 ∂θ1 L12 ðwÞ ¼  c24

∂4 w ∂4 w ∂3 w ∂2 w ∂4 w ∂3 w  ðc14 þ c23 þ 2c32 Þ þ ð3c14 þ c23 þ 4c32 Þ  2ðc14 þ c32 þc24 Þ 2 c13 4 þð4c13 þ c23  c14 Þ 3 4 2 2 2 ∂z ∂z ∂θ1 ∂z ∂θ1 ∂z∂θ1 ∂θ1

∂2 w ∂w  ð5c13 þ 3c23  3c14  c24 Þ 2 þ 2ðc13 þ c23  c14  c24 Þ ∂z ∂z !
! 
2  2 2 2 ∂ Ψ 1 ∂Ψ 1 ∂ w ∂w ∂Ψ 1 ∂ w ∂w 2z ∂ Ψ 1 þ e þ 2Ψ þ 2Ψ þ  þ 3 þ L13 ðΨ 1 ; wÞ ¼ e2z 1 1 ∂z ∂z ∂z2 ∂z ∂z2 ∂θ21 ∂θ21 ∂z !
2 
2  ∂ Ψ 1 ∂Ψ 1 ∂ w ∂w ∂2 w ∂2 w þ  þ þS41 e4z K w w  K p S21 e2z  2e2z ∂z∂θ1 ∂θ1 ∂z∂θ1 ∂θ1 ∂z2 ∂θ21 ∂4 Ψ 1 ∂3 Ψ 1 ∂2 Ψ 1 þ ðb21  b12  4b22 Þe2z 3 þ ð5b22 þ3b12  3b21 b11 Þe2z 2 ∂z4 ∂z ∂z 4Ψ 1 ∂Ψ ∂ 1 þ ð2b31 þ b21 þb12 Þe2z þ 2ðb11 þ b21  b12  b22 Þe2z þ ∂z ∂z2 ∂θ21

L21 ðΨ 1 Þ ¼ b22 e2z

ðA:1Þ

188

A.H. Sofiyev / Thin-Walled Structures 80 (2014) 178–191

 ð4b31 þ 3b12 þ b21 Þe2z L22 ðwÞ ¼  b14

∂3 Ψ 1 ∂2 Ψ 1 ∂4 Ψ 1 þ 2ðb31 þ b21 þb11 Þe2z 2 þ b11 e2z 4 2 ∂z∂θ1 ∂θ1 ∂θ1

∂4 w ∂4 w ∂3 w þ ð2b32  b13  b24 Þ þ ð3b24  4b32 þ b13 Þ þ 2ðb32  b24  b14 Þ 2 4 2 ∂θ1 ∂z ∂θ1 ∂z∂θ21


2  ∂2 w ∂4 w ∂3 w ∂2 w ∂w ∂ w ∂w S1 ez cot γ  b23 4 þ ðb13 b24 þ 4b23 Þ 3 þ ð3b24  3b13  5b23 þ b14 Þ 2 þ 2ðb13 þ b23  b14  b24 Þ þ  2 2 ∂z ∂z ∂z ∂z ∂z ∂z ∂θ1
2

2 2
 ∂w ∂w ∂2 w ∂w ∂2 w ∂2 w ∂ w ∂w ∂2 w ∂w  2  þ2    L23 ðw; wÞ ¼  ∂θ1 ∂θ1 ∂z∂θ1 ∂z ∂z ∂z∂θ1 ∂z ∂z2 ∂z ∂θ21 

ðA:2Þ

Appendix B In Eq. (29), Δ00j ; j ¼ 1; 2; …; 15 are parameters and given as Δ001 ¼ Δ011 þ Δ012 þ Δ013 þ Δ014 þ Δ015 ;

Δ002 ¼ Δ021 þ Δ022 þΔ023 þ Δ024

ðB:1Þ

where Δ011 ¼

β2m ðe2z0  1Þ 4 4ð1 þβ2m Þ sin

γ

  ð  2δ14 þ 3δ13 þδ15  δ13 β2m Þð1 þβ2m Þ cos 4 γ þ  6δ13 þ ðδ10  δ12 Þβ2n sin 2 γ:


 1  2δ15 þ 4δ14  2β2m 2δ13  2δ14 þ δ15  δ10 β2n sin 2 γ þ 2δ13 β4m cos 2 γ þ 3δ13  δ9 β4n sin 4 γ 2

2 2 2 þ ðδ12  δ10 Þβn sin γ 2δ14 þδ15 þ βm ðδ15  δ10 β2n sin 2 γ þ2δ13 2δ14 Þ  δ13 β4m

Δ012 ¼

A1 β2m ðe2z0  1Þ  4

4ð1 þβ2m Þ sin γ

½2δ4 5δ1 þ 4δ2  3δ3 þ β2m ðδ3 þ 10δ1  4δ2 Þ  β4m δ1  cos 4 γ  8δ2 þ 6δ3

þ ½ð2δ7  3δ6  δ8 Þβ2n sin 2 γ  4δ4 þ 10δ1 þ β2m ðδ6 β2n sin 2 γ  2δ3  20δ1 þ 8δ2 Þ þ 2β4m δ1  cos 2 γ  δ5 β4n sin 4 γ þ ð3δ6  2δ7 þ δ8 Þβ2n sin 2 γ  β2m ðδ6 β2n sin 2 γ þ 4δ2  δ3  10δ1 Þ A1 β2m ð3 þ 2β2m ÞS1 ðe3z0  1Þcot γ  5δ1 þ 2δ4 þ 4δ2  3δ3 β4m δ1 þ 3ð9 þ4β2m Þ Δ013 ¼

A2 βm ð1  e2z0 Þ 4ð1 þβ2m Þ sin 4 γ

 ½δ3  δ4 þ δ1  δ2  β2m ð  6δ2 þ 3δ3  δ4 þ 10δ1 Þ þ β4m ð5δ1  δ2 Þ cos 4 γ

þ 〈ðδ6  δ7 þ δ8 Þβ2n sin 2 γ  2δ3 þ 2δ4 þ2δ2  2δ1  β2m ½ð3δ6  δ7 Þβ2n sin 2 γ þ 2δ4  20δ1 þ 12δ2  6δ3  4

2

 2β4m ð5δ1  δ2 Þ〉 cos 2 γ þ δ5 β4n sin γ þðδ7  δ8  δ6 Þβ2n sin γ þ δ3 δ4 þ δ1 δ2

A2 β3m S1 ðe3z0  1Þcot γ 2 þ β2m ½ðδ6  δ7 Þβ2n sin γ  3δ3 þ 6δ2 þ 3δ4  10δ1  þ β4m ð5δ1  δ2 Þ þ 3ð9 þ 4β2m Þ Δ014 ¼

A92 βm S1 cot γ



〈 β2m ½6δ3  2β2m ð3δ1  2δ2 Þ  4δ4  cos 4 γ 6ð9 þ4β2m Þ sin 4 γ þ f4β4m ð3δ1  2δ2 Þ 2β2m ½ð2δ7  3δ6 Þβ2n sin 2 γ  4δ4 þ 6δ3   6β2n  2β4m ð3δ1  2δ2 Þ þ 2β2m ½ð2δ7  3δ6 Þβ2n sin 2 γ  2δ4 þ 3δ3  þ6β2n 3  3ð2:25 þ β2m Þ sin γ cos γS1 ðe4z0  1Þ〉

Δ015 ¼ 

sin 2 γδ8 g cos 2 γ sin 2 γð  δ5 β2n sin 2 γ þ δ8 Þgðe3z0  1Þ

A102 S1 β2m cot γ n 2 2 〈 ½3δ4 þ β2m ð2δ3 3δ2 Þ β4m δ1  cos 4 γ þ ½ð3δ7  2δ8 Þβ2n sin γ  6δ4  4β2m ðδ3  δ6 β2n sin γ  3δ2 Þ þ 4β4m δ1  cos 2 γ 3ð9 þ 4β2m Þ sin 4 γ o  2δ5 β4n sin 4 γ þ ð2δ8  3δ7 Þβ2n sin 2 γ þ 3δ4 þ β2m ð2δ3  2δ6 β2n sin 2 γ 3δ2 Þ  2β4m δ1 ðe3z0  1Þ 3 þ ð9 þ 4β2m Þ sin 3 γ cos γS1 ðe4z0  1Þ〉 8

Δ021 ¼

A91 βm S1 ð1  e4z0 Þcot γ A91 βm ðe3z0  1Þ 2 þ 〈βm ½3δ3  β2m ð3δ1 2δ2 Þ 2δ4  cos 4 γ 8 3ð9 þ 4β2m Þ sin 4 γ þ f2β4m ð3δ1  2δ2 Þ β2m ½ð2δ7 3δ6 Þβ2n sin 2 γ  4δ4 þ 6δ3   2β2n δ8 sin 2 γg cos 2 γ 2

2

2

 β4m ð3δ1  2δ2 Þ þ β2m ½ð2δ7  3δ6 Þβ2n sin γ  2δ4 þ 3δ3  þ3β2n ðδ8  δ5 β2n sin γÞ sin γ〉 Δ022 ¼

A101 β2m S1 ð1  e4z0 Þcot γ A101 β2m ðe3z0  1Þ  4 þ ½4βm δ1  6δ4 2β2m ð2δ3  3δ2 Þ cos 4 γ 4 3ð9 þ 4β2m Þ sin 4 γ þ ½4ðδ8  2δ7 Þβ2n sin 2 γ þ 12δ4 4β2m ð3δ2  2δ3 þ δ6 β2n sin 2 γÞ  8β4m δ1  cos 2 γ þ 4δ5 β4n sin 4 γ þ ð6δ7  4δ8 Þβ2n sin 2 γ 6δ4 þ 2β2m ð3δ2  2δ3 þ 2δ6 β2n sin 2 γÞ þ 4β4m δ1 g

A.H. Sofiyev / Thin-Walled Structures 80 (2014) 178–191

Δ023 ¼

189

A11 βm ð4 þ 7β2m þ 9β4m Þðe4z0 1ÞS1 cot γ 9A11 βm ð1  e3z0 Þ  〈9½6δ4  9δ3 þ β2m ð81δ1 þ 8δ3 54δ2 Þ  72β4m δ1 β2m cos 4 γ þ 2 2 4 8ð1 þ βm Þð4 þ βm Þ ð9 þ 16β2m Þð9 þ4β2m Þ sin γ 2

2

þfβ2n δ8 sin γ þ β2m ½ð54δ7  81δ6  8δ8 Þβ2n sin γ þ162δ3  118δ4 þ 96β6m δ1 4β4m ð81δ1 18δ2 þ8δ3  4δ6 β2n sin 2 γÞ=3g cos 2 γ 9β2n ðδ8  δ5 β2n sin 2 γÞ sin 2 γ β2m ½8δ5 β4n sin 4 γ þ ð54δ7 81δ6  8δ8 Þβ2n sin 2 γ  54δ4 þ 81δ3  þ β4m ð8δ3  8δ6 β2n sin 2 γ þ 81δ1  54δ2 Þ  72β6m δ1 〉 Δ024 ¼

3A12 β2m S1 ð1  e4z0 Þð2 þ 3β2m Þcot γ 3A12 β2m ðe3z0 1Þ þ 〈½β2m ð16δ4 þ 81δ2  54δ3 Þ 4 4 ð9 þ 16β2m Þð9 þ 4β2m Þ sin γ 2

81δ4 162β4m ð4δ2  27δ1 Þ cos 4 γ þ f18β2m ½ð2δ7  27δ6 Þβ2n sin γ þ 54δ3  8δ4  81δ2  þð54δ1  81δ7 Þβ2n sin 2 γ þ162δ4 þ 324β4m ð4δ2  27δ1 Þg cos 2 γ þ 54δ5 β4n sin 4 γ þ 27ð3δ7  2δ8 Þ β2n sin 2 γ  81δ4  9β2m ½ð8δ7 54δ6 Þβ2n sin 2 γ þ 54δ3  8δ4  81δ2   162β4m ð4δ2 27δ1 Þ〉: and Δ003 ¼

f½4A1 β3m ð8β2n  3Þ þ 2A2 ð8β4m  7β2m  9 þ 9β2n þ4β2m β2n Þβm  A16 ð3 þ 2β2m Þð9 þ 16β2m Þgβ2m ðe3z0  1Þ 3ð9 þ 4β2m Þð9 þ 16β2m Þ 

Δ004 ¼ 

½2A3 ð8β4m þ 66β2m þ27 þ8β4m β2n Þ A4 βm ð8β4m þ 16β2m þ9 þ 9β2n þ 28β2m β2n Þβ2m ðe3z0 1Þ 6ð64β4m þ 180β2m þ 81Þ ½2A51 ðβ2m þ 3 þ β2n þ β2m β2n Þ þ A61 βm ðβ2m þ 5 þ 3β2n Þβ2m ðe4z0  1Þ 8ð4 þβ2m Þ

þ

½2A91 ðβ4m þ β2m  3 þ β2m β2n þ 4β2n Þ þ A101 βm ð  β2m þ 5 þ 4β2m β2n  2β2n Þβ3m ðe4z0  1Þ 32ð4 þβ2m Þð1 þ β2m Þ



A11 β3m ð27β2m 12β4m þ 12  27β6m  16β2n  40β2m β2n  39β4m β2n Þðe4z0  1Þ 256 þ 896β2m þ 784β4m þ 144β6m

þ

A12 β4m ð195β2m þ 135β4m þ60 24β2n  30β2m β2n  36β4m β2n Þðe4z0  1Þ 2ð256 þ 896β2m þ 784β4m þ144β6m Þ



β2m A13 ð12 þ 43β2m þ43β4m þ β6m þ 4β2n þ 29β2m β2n þ 70β4m β2n Þðe4z0 1Þ 64 þ224β2m þ 196β4m þ 36β6m



β3m A14 ð20 þ 75β2m þ85β4m þ 12β2n þ 33β2m β2n  24β4m β2n Þðe4z0  1Þ : 64 þ224β2m þ 196β4m þ 36β6m

( β2m ðe4z0 1Þ 2A7 ðβ4m þ3 þ 4β2m  7β2n  10β2m β2n Þ þ A8 βm ðβ4m þ 5 þ 6β2m þ 3β2n þ 9β2m β2n Þ Δ005 ¼ 8 2ð4 þ 5β2m þ β4m Þ ) 2 2 2 2 2 A15 ðβm þ3  7βn  2βm βn Þ  2A52 ðβm þ3 þ β2n þ β2m þ β2m β2n Þ  A62 βm ðβ2m þ 5 þ 3β2n Þ þ 4 þ β2m Δ006 ¼

( β2m S1 ðe4z0  1Þcot γ 2A53 ðβ2m þ 3 þ β2n þ β2m β2n Þ  A63 βm ðβ2m þ 5 þ 3β2n Þ 8 4 þ β2m þ

) 2A92 βm ðβ4m þ β2m  3 þ β2m β2n þ4β2n Þ þ A102 β2m ð  β2m þ 5 þ 4β2m β2n  2β2n Þ 4ð4 þ β2m Þð1 þ β2m Þ

Δ007 ¼

S21 β4m ð1 þ 4β2m Þðe4z0  1Þ ; 16ðβ2m þ 1Þðβ2m þ 4Þ

Δ009 ¼

ð3δ13 δ15 þ 4δ13 β2m Þð1  e2z0 Þβ4m þ A16 ðδ3 δ4 þ δ1  δ2 Þðe2z0  1Þβ2m 2ðβ2m þ 1Þ

Δ008 ¼ 

S31 β2m cot γðe4z0  1Þ ; 4ðβ2m þ 4Þ



A3 β4m ½δ4 þ δ2  δ1  δ3  ð8δ4  18δ3 þ 32δ2  25δ1 Þ  8ð20δ1 þ δ3  3δ2 Þβ4m þ 32δ1 β6m ðe2z0  1Þ 2ð1 þ 4β2m Þðβ2m þ 1Þ

þ

128A3 β6m ðe3z0  1ÞS1 cot γ 56A4 β3m ðe3z0  1ÞS1 cot γ þ 3ð4β2m þ 9Þð16β2m þ 9Þ 3ð16β2m þ 9Þ

þ

A4 β3m ½9δ2 7δ3  11δ1 þ 5δ4 þ ð30δ1  56δ2  4δ4 þ 20δ3 Þβ2m 28ð7δ1  δ2 Þβ4m ðe2z0  1Þ 2ð1 þ 4β2m Þðβ2m þ1Þ

þ

16A53 β4m ½9δ4  9δ3  32β4m δ1 þ4ð2δ3  9δ2 þ9δ1 Þβ2m ðe3z0  1ÞS1 cot γ

3ð16β2m þ 9Þð4β2m þ 9Þ 3 2A63 βm ½16ð2δ4  9δ3 þ9δ2 Þβ2m þ 64ð9δ1 2δ2 Þβ4m  28δ14 ðe3z0  3ð4β2m þ 9Þð16β2m þ 9Þ

þ

A53 β2m ðβ2m þ 1Þðe4z0  1ÞS21 cot2 γ

 1ÞS1 cot γ

2ðβ2m þ4Þ þ

3A63 β3m ðe4z0  1ÞS21 cot2 γ 4ðβ2m þ 4Þ

ðB:2Þ

190

Δ010 ¼

A.H. Sofiyev / Thin-Walled Structures 80 (2014) 178–191

16A51 β4m ðe3z0  1Þ½9δ4  9δ3  32β4m δ1 þ4ð2δ3  9δ2 þ9δ1 Þβ2m  A51 β2m ðβ2m þ 1Þðe4z0  1ÞS1 cot γ þ 3ð16β2m þ 9Þð9β2m þ 4Þ 2ðβ2m þ 4Þ 

2A61 β3m ðe3z0  1Þ½16ð2δ4  9δ3 þ 9δ2 Þβ2m  36δ4 þ 64ð9δ1  2δ2 Þβ4m  3A61 β3m ðe4z0  1ÞS1 cot γ þ 3ð16β2m þ 9Þð9β2m þ 4Þ 4ð9β2m þ 4Þ

þ

256A13 β4m ðe3z0  1Þ½64β4m ðδ2  11δ1 Þ þ 4β2m ð36δ1 þ 11δ3  36δ2  δ4 Þ  9δ3 þ9δ4  3ð9β2m þ 4Þð1 þ 4β2m Þð16β2m þ9Þ

þ

A13 β2m ð4 þ 29β2m þ 70β4m Þðe4z0  1ÞS1 cot γ 3A14 β3m ð8β4m  4  11β2m Þðe4z0  1ÞS1 cot γ 

2ð9β2m þ4Þðβ2m þ4Þðβ2m þ 1Þ 2ð9β2m þ 4Þðβ2m þ 4Þðβ2m þ 1Þ

þ

16A14 β3m ½1024δ1 β6m þ 64β4m ð11δ2  δ3  36δ1 Þ  β2m ð11δ4 þ 36δ2  36δ3 Þ þ9δ14 ðe3z0  1Þ 3ð4β4m þ 9Þð1 þ 4β2m Þð16β2m þ9Þ

Δ0011 ¼

½4A1 β3m ð1 þ 8β2n Þ þA2 ð9 þ 19β2m þ 8β4m  9β2n  7β2m β2n β3m ðe3z0  1Þ ½2A92 ð5 þ β2m  3β2n ÞÞ A102 βm ð1 þ 4β2n Þðe4z0  1Þβ3m S1 cot γ  ; 3ð81 þ180β2m þ 64β4m Þ 32ðβ2m þ 4Þ

Δ0012 ¼ 

½2A51 ð6β2m þ 1Þ þ 5A61 βm β4m ðe4z0  1Þ ½A13 ð4 þ 37β2m þ 108β4m Þ  βm A14 ð  20  59β2m þ 36β4m Þβ4m ðe4z0  1Þ  ; 4ð9β2m þ 4Þðβ2m þ4Þ 8ð9β2m þ 4Þðβ2m þ 4Þðβ2m þ1Þ

 A52 β2m 32ðβ2m þ 4Þ½  32β4m δ1 þ 4ð2δ3  9δ2 þ 9δ1 Þβ2m 2 2 2 6ð4βm þ9Þð16βm þ 9Þðβm þ 4Þ þ 9δ4  9δ3 β2m ðe3z0  1Þ þ 3ð16β2m þ 9Þð4β2m þ 9Þðβ2m þ 1Þðe4z0  1ÞS1 cot γ  2 4A62 β3m  2ðβm þ 4Þ½4ð2δ14 9δ13 þ9δ12 Þβ2m þ288ð9δ1  2δ2 Þβ4m  162δ4 ðe3z0 2 3ð4βm þ9Þð16β2m þ 9Þðβ2m þ 4Þ

Δ0013 ¼

1Þ ð4β2m þ 9Þðe4z0  1ÞS1 cot γ ;

½2A52 ð6β2m þ 1Þ þ 5A62 βm β4m ðe4z0  1Þ ½2A91 ð5 þβ2m 3β2n Þ þ A101 βm ð1 þ4β2n Þβ3m ðe4z0  1Þ þ 4ð9β2m þ 4Þðβ2m þ4Þ 32ðβ2m þ 4Þ 4z0  ðe  1Þ þ 2A11 β3m ½ð45β2m þ 27β4m þ 45Þð12 þ 27β2m Þ 32ð9β2m þ 4Þðβ2m þ4Þðβ2m þ 1Þ

Δ0014 ¼ 

 12β2n  27β2m β2n þ 75β4m β2n  þ 3A12 β4m ð21β2m þ 27β4m þ 4 þ 16β2n þ 64β2m β2n  12β4m β2n Þg; Δ0015 ¼ 

½2A53 ð6β2m þ 1Þ þ 5A63 βm ðe4z0 1Þβ4m S1 cot γ 64A16 β6m ðe3z0 1Þ  2 2 4ð9βm þ 4Þðβm þ 4Þ 3ð4β2m þ 9Þð16β2m þ 9Þ

ðB:3Þ

where δ1 ¼ c12 ; δ2 ¼ c11  c22 þ 4c12 ; δ3 ¼ 5c12 þ 3c11 3c22  c21 ; δ4 ¼ 2ðc11  c22 þc12  c21 Þ; δ5 ¼ c21 ; δ6 ¼ c11 2c31 þ c22 ; δ7 ¼ c11  4c31 þ 3c22 ; δ8 ¼ 2ðc22  c31 þ c21 Þ; δ9 ¼ c24 ; δ10 ¼ c14 þ c23 þ 2c32 ; δ11 ¼ 3c14 þ c23 þ 4c32 ; δ12 ¼ 2ðc14 þ c32 þ c24 Þ; δ13 ¼ c13 ; δ14 ¼ 4c13 þ c23  c14 ; δ15 ¼ 5c13 þ 3c23  3c14  c24 ; δ16 ¼ 2ðc13 þ c23  c14  c24 Þ;

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ðB:4Þ

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