Buckling analysis of freely-supported functionally graded truncated conical shells under external pressures

Accepted Manuscript BUCKLING ANALYSIS OF FREELY-SUPPORTED FUNCTIONALLY GRADED TRUNCATED CONICAL SHELLS UNDER EXTERNAL PRESSURES A.H. Sofiyev PII: DOI: Reference:

S0263-8223(15)00490-0 http://dx.doi.org/10.1016/j.compstruct.2015.06.026 COST 6525

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Composite Structures

Please cite this article as: Sofiyev, A.H., BUCKLING ANALYSIS OF FREELY-SUPPORTED FUNCTIONALLY GRADED TRUNCATED CONICAL SHELLS UNDER EXTERNAL PRESSURES, Composite Structures (2015), doi: http://dx.doi.org/10.1016/j.compstruct.2015.06.026

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BUCKLING ANALYSIS OF FREELY-SUPPORTED FUNCTIONALLY GRADED TRUNCATED CONICAL SHELLS UNDER EXTERNAL PRESSURES

A.H. Sofiyev*

Department of Civil Engineering of Engineering Faculty of Suleyman Demirel University, Isparta, Turkey

TOTAL NUMBER OF PAGES: 46, FIGURES: 7, TABLES: 3

Corresponding author:

Prof. Dr. Abdullah SOFIYEV Department of Civil Engineering of Engineering Faculty of Suleyman Demirel University, 32260 Isparta, Turkey E-mail: [email protected] Tel: 0090 246 211 1195 Fax: 0090 246 237 0859

Abstract

This article presents a method to study the buckling of freely-supported functionally graded (FG) truncated and complete conical shells under external pressures in the framework of the shear deformation theory (SDT). The basic relations, modified Donnell type stability and compatibility equations have been obtained on the basis of SDT. The material properties of truncated conical shells are functionally graded in the thickness direction according to a volume fraction power law distribution. To solve this problem is used an unknown parameter λ in the approximation functions. One of innovations is to achieve closed-form solutions for critical lateral and hydrostatic pressures of freely-supported FG truncated and complete conical shells in the framework of the SDT. The parameter λ which is included in the obtained expressions is get from the minimum conditions of critical external pressures. Finally, influences of shear stresses, volume fraction index and shell characteristics on the critical lateral and hydrostatic pressures are investigated.

Keywords: Buckling; FG truncated and complete conical shells; external pressures; critical lateral and hydrostatic pressures; shear deformation theory

1. Introduction

The structural elements in the form of circular conical shells for several decades are the subject of numerous and diverse research. Interest to problems of buckling of conical shells under an external pressure is due primarily to the fact that they are the main components of load-bearing constructions which are used in aviation and rocketry, underwater vehicles, missiles, tanks, pressure vessels and piping and buildings of modern power plants. The buckling behaviors of conical shells under external pressures have been subject of early studies based on the classical shell theory (CST). Singer [1] presented buckling of circular conical shells under an external pressure. Agenesov and Sachenkov [2] studied the stability and vibration of circular conical and cylindrical shells subjected to an external pressure at different boundary conditions. Thurtson [3] investigated the effect of boundary conditions on the buckling of conical shells under a hydrostatic pressure. Baruch et al. [4] studied the buckling of an isotropic conical shell under a hydrostatic pressure using the Donnell type shell theory. More detailed descriptions on the development of researches about buckling of homogeneous isotropic conical shells under external pressures based on the CST may be found in works [5-9]. Functionally graded materials (FGMs) are heterogeneous materials in which the material properties are varied continuously from point to point. Typically, FGMs are made from a mixture of ceramic and metal or a combination of different materials. The concept of FGMs was first proposed in 1984 by a group of Japanese materials scientists [10,11]. Initially, FGMs were designed as thermal barrier for aerospace application and fusion reactors. Later on, FGMs were developed for military, automotive, biomedical application, semi-conductor industry and etc. The basic knowledge on the design, manufacture, applications and techniques of FGMs may be found in the studies of Suresh and Mortensen [12], and Ichikawa

[13]. A comprehensive survey of the relevant theoretical methodologies and modeling of FGM structures can be found in the literature [14-16]. More detailed descriptions on the FGMs and the behaviors of shells made of FGMs can be found in Refs. [17, 18]. In recent years, FG truncated conical shells are being used as structural components in modern industries of different engineering fields such as aerospace, mechanical and nuclear engineering. In order to use them effectively and successfully, a good understanding of the buckling of FG conical shells under external pressures is necessary. It is noted that the buckling analysis of FG conical shells is more complicated than that of homogeneous material structures owing to the spatial variations of material properties. The inherent complexity for solving the equations of buckling of conical shells thus the closed-form solutions is restricted. The first studies on the solution of the buckling problem of FG truncated conical shells under external pressure are given by the classical shell theory (CST). For example, Naj et al. [19] studied thermal and mechanical instability of FG truncated conical shells based on CST. Sofiyev [20] presented the vibration and buckling behaviors of FGM conical shells subjected to external pressures on the basis of the CST. Torabi et al. [21] analyzed linear thermal buckling behavior of truncated hybrid FGM conical shells in the framework of the CST. Sofiyev and Kuruoglu [22] presented the solution of the buckling problem of functionally graded truncated conical shells subjected to external pressure with the mixed boundary conditions based on the CST. Since the shear stresses effect is neglected, the CST cannot be applied to moderately thick shells. Shear deformation theories eliminate deficiency of CST by accounting for transverse shear stresses of FG conical shells. Extensive investigations have been carried out to study the eigenvalue problems of FG conical shells based on two dimensional (2-D) theories, such as CST and SDT. Most of these studies are devoted to the vibration problems of FG conical shells and have been limited to numerical solutions [23-32].

In recent years, many studies have appeared in the literature in connection with the problems of buckling of FG cylindrical shells under external pressures in the framework of the SDT. Shen [33] analyzed the post-buckling behavior of functionally graded cylindrical shells under lateral pressure in thermal environments using the boundary layer shell theory based on the SDT. Shen and Noda [34] studied the postbuckling of FGM cylindrical shells under combined axial and radial mechanical loads in thermal environments in the framework of the SDT. Shen [35] presented a study on the postbuckling response of a shear deformable functionally graded cylindrical shell of finite length embedded in a large outer elastic medium and subjected to axial compressive loads in thermal environments. Khazaeinejad et al. [36] investigated the buckling of functionally graded cylindrical shells under combined external pressure and axial compression on the basis of SDT. Lang and Tran et al. [37] analyzed isogeometric of functionally graded plates using higher-order shear deformation theory. Wu et al. [38] presented the three-dimensional (3D) linear buckling analysis of simply-supported, multilayered functionally graded material (FGM) circular hollow cylinders under combined axial compression and external pressure using unified formulations of finite cylindrical layer methods based on the Reissner mixed variational theorem and the principle of virtual displacements. Lang and Xuewu [39] analyzed buckling and vibration behaviors of functionally graded magneto-electro-thermo-elastic circular cylindrical shell under external pressure using Hamilton principle and higher order shear deformation theory. Neves, et al. [40] investigated the static, free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique. Sofiyev and Kuruoglu [41] examined buckling and vibration of shear deformable functionally graded orthotropic cylindrical shells under external pressures. Dai et al. [42] analyzed the buckling for a ring-stiffened FGM cylindrical shell under hydrostatic pressure and thermal loads in the framework of the SDT. However, investigations on the

buckling behaviors of FG conical shells under external pressures in the framework of the STD are limited in number. Bhangale et al. [43] developed a finite element formulation based on the FSDT to study the thermal buckling and vibration behavior of truncated conical shells in a high-temperature environment. Asemi et al. [44] studied the elastic behavior of functionally graded thick truncated cone under hydrostatic internal pressure using finite element method based on Rayleigh–Ritz energy formulation. Sofiyev [45] examined the vibration and stability of shear deformable FGM truncated conical shells subjected to an axial load. After a close review of the literature, we found that there is no work devoted to a closedform solution of buckling of FG truncated and complete conical shells under lateral and hydrostatic pressures in the framework of the SDT. The purpose of this paper is to present a method to study the FG truncated and complete conical shells under external pressures in the framework of the SDT. The novelty of this study is to obtain the closed-form solutions for the critical lateral and hydrostatic pressures of FG conical shells in the framework of the SDT which caters for both thin and relatively thick FG truncated and complete conical shells.

2. Formulation of the problem

A functionally graded truncated conical shell is shown in Fig. 1, wherein a coordinate system OSz is established on the middle surface of the conical shell. The coordinate S is measured along the cone generator with origin at the boundary of the small base, the angle  is the circumferential coordinate and z is the thickness coordinate. The geometric parameters of FG conical shell are represented by length L , semi-vertex angle  , thickness h , the radii at the two ends R1 and R2 ( R2  R1 ) and the distances from vertex to the small and large edges S1 and S 2 , respectively.

The FG truncated conical shell is subjected to lateral and hydrostatic pressures:

N S0  0.5SP1 tan , N 0  SP2 tan ,

where N S0 , N0

N S0  0

(1)

and N S0 are the membrane forces for the condition with zero initial

moments. As P1  0, P2  PL and P1  P2  PH , the external pressures turn into the lateral and hydrostatic pressures, respectively. The conical shell material assumed in the following is a functionally graded linear elastic one. The volume fraction of ceramics, Vc (Z ) , varied depending on the normalized thickness coordinate as a power-law function which can be expressed as [14]:

Vc ( Z )  Z  0.5k , Vc (Z )  Vm (Z )  1

(2)

where Vm (Z ) is the volume fraction of metal and k is the volume fraction index which describes the material distribution along with the thickness direction of the FG shell and takes only positive values 0  k   . The subscripts c and m correspond to the ceramic and metallic constituents, respectively. Based on the above definition, it follows that the outer surface of the conical shell will be metal-rich. Based on the volume fraction definition and law of mixtures, the effective material properties of the FG truncated conical shells are defined as [14]:

E FG (Z )  ( Ec  Em )Z  0.5k  Em ,

 FG (Z )  ( c  m )Z  0.5k   m ,

(3)

where E m , ν m and E c , ν c are the Young’s modulus and Poisson’s ratio of the metal and ceramic surfaces of the FG truncated conical shell, respectively. From Eq. (3), it is clear that the shells fully ceramic with k  0 and fully metal with k   . The detailed descriptions of the material variation profiles for FGMs are presented by Tornabene [16].

3. Governing equations

The stress-strain relations of FG truncated conical shells in the framework of the STD are given in the following form [45,50]:

 S   B11 B12 0   S      B B       12 11 0      S   0 0 B66   S 

(4)  S   B55 0   S          0 B44    

where  S and   are the normal stresses,  S is the in-plane shear stress,  S and   are the thickness shear stresses.  S and   are the normal strains,  S is in-plane shear strain and

 S and   are the thickness shear strains and Bij (i, j  1,2,4,5,6) are the stiffness coefficients depending on the normalized thickness coordinate Z and are:

B11 

B12

( Ec  E m )Z  0.5k  E m



1  ( c   m )Z  0.5k   m

( 

c





2

,

  m )Z  0.5k   m ( Ec  E m )Z  0.5k  E m



1  ( c   m )Z  0.5k   m

B44  B55  B66 



( Ec  E m )Z  0.5k  E m





2

2  2 ( c   m )Z  0.5k   m

(5)



The shear stresses of FG conical shells in the framework of the STD vary depending on the thickness coordinate as [45-50]:

 df1 ( z )   S   dz 1 ( S ,  )         df 2 ( z )  ( S , ) 2  dz 

(6)

where 1 ( S ,  ) and  2 ( S ,  ) are the rotations of normal's to the middle surface with the respect to the  and S axis, respectively, and f i ( z ), (i  1,2) are the posteriori specified shape functions of the shear stresses,  S and   , and distributed a sinus hyperbolic manner across the thickness of the FG conical shell, i.e., f1 z   f 2 z   hsinhZ   zcosh1 / 2 . The expressions for the strains of an FG conical shell in the framework of the SDT can be expressed as:

 2w     S   0 S   2   1    0      S       S      1  2 w 1 w    1  2        0   z  2    I1 0   I2        S  2 S S   1    S   1       2     2   S   0 S   1  w  1 w   S      S   S S S 2  

(7)

where    sin  and  0 S ,  0 ,  0 S are strains on the reference surface of FG conical shells and I j ( j  1,2 ) are functions and defined as

coshZ   cosh1 / 2 dz, B55 ( Z ) 0 z

I1  

coshZ   cosh1 / 2 dz B44 ( Z ) 0 z

I2  

(8)

Following the virtual work principle, five equations for static buckling of the truncated conical shell element in terms of internal actions may be written in the following manner [51]:

N S N S  N 1 N S   0 S S S 

(9)

1 N N S 2 N S   0 S  S S

QS QS Q N 2w N    cot   N S 0 2   0 S S S S S S

(10)

 1  2 w w     S  2  S   0  

(11)

M S M S  M  1 M S    QS  0 S S S 

(12)

M S 1 M  2M S    Q  0 S S  S

(13)

where

N S , N  and N S are the in-plane meridional, circumferential and shearing force

resultants, M S , M  and M S are the analogous couples, while QS and Q are the transverse shear force resultants and expressed as [46]:

N S , M S  N , M   h / 2      h / 2  N S , M S 

 S    1, z  d z     S 

(14) Q S  h / 2 Q       h / 2

 Sz   

 dz 

The stress resultants can be represented in the following manner in terms of the undetermined Airy stress function,  ( S ,  ) [51]:

 1  2 1   N S   2 2 S S    S      2  N    2    S    2  N S   1    1   S S S 2 

        

(15)

The Eqs. (9) and (10) are satisfied identically by setting (15). The compatibility equation on the reference surface of truncated conical shells is as follows [45, 51]:

cot   2 w 1  2 0 S 1  0 S  2 0 1  2 0 S 2  0 1  0 S       0 S S 2 S S S S S S S 2  S 2 S 2  2

(16)

Using Eqs. (4), (7) and (14), we can get the force and moment resultants and strains on the reference surface in terms of the  , w, 1 ,  2 and then substituting resulting expressions with the relations (15) into the Eqs. (10)-(12), we obtain after some mathematical manipulations, the following set of basic equations:

 L11    L21    L31   L  41

L12 L13 L14    0         L22 L23 L24   w  0          L32 L33 L34  1  0         L42 L43 L44   2  0

(17)

where Lij (i, j  1,2,...,4) , are differential operators and given in the Appendix A. The set of Eqs. (17) are the buckling and compatibility equations of FG conical shells under external pressures in the framework of the SDT.

4. Solution of governing equations

The functionally graded truncated conical shell is assumed to be freely supported, so the equations describing the freely supported boundary conditions can be written as follows [45]:

w  0, M S  0,  2  0,  ,  0 as x   x 0 and x  0

The functions satisfying the boundary conditions have the following forms [45]:

(18)

 ( x,  )   mn S 2 e (  1) x sin ( m1 x) cos( n1 ) w( x,  )  f mn e x sin ( m1 x) cos( n1 )

(19)

1 ( x, )  1mn e x cos ( m1 x) cos( n1 )  2 ( x, )   2 mn e x sin ( m1 x) sin( n1 )

where  mn , f mn , 1mn ,  2mn are unknown amplitudes,  is a unknown parameter that will be determined from the minimum conditions of critical lateral or hydrostatic pressures and

m1 

S m n S , n1  , x  ln , x0  ln 2 , in which, m is the half wave number in the axial x0 sin  S2 S1

direction and n is the circumferential wave number. Substituting (19) into set of Eqs. (17), and multiplying each equation by the corresponding eigenfunction, then integrating over the domain of solution, we obtain after some mathematical manipulations, the following system equations in matrix form:

U 11  U 12   U 21  U 22   U  U 32  31   U 41  U P1 P1  U P2 P2

U 13 U 14   mn  0         U 23 U 24   f mn  0        U 33 U 34  1mn  0          U 43 U 44   2 mn  0

(20)

where U ij ( i , j  1,2,...,4 ) , are the FG conical shell parameters, U P1 and U P2 are the external loading parameters which defined in the Appendix B . For the non-trivial solution, the determinant of set of Eqs. (20) must be zero, i.e.:

U 11

 U 12

U 13 U 14

U 21

 U 22

U 23 U 24 0

 U 32

U 31

(21)

U 33 U 34

U 41 (  U P1 P1  U P2 P2 ) U 43 U 44

The expansion of the determinant (21) is as follows





U 411   P1U P1  P2U P2  2  U 43 3  U 44  4  0

(22)

where  j ( j  1,2,3,4) is expansion of the determinant of a square matrix and the following definitions apply:

 1  U 12U 23U 34  U 12U 24U 33  U 22U 33U 14  U 22U 13U 34  U 32U 13U 24  U 32U 23U 14  2  U 11U 23U 34  U 11U 24U 33  U 21U 33U 14  U 21U 13U 34  U 31U 13U 24  U 31U 23U 14 (23)

 3  U 11U 22U 34  U 11U 24U 32  U 21U 32U 14  U 21U 12U 34  U 31U 12U 24  U 31U 22U 14  4  U 11U 22U 33  U 11U 23U 32  U 21U 32U 13  U 21U 12U 33  U 31U 12U 23  U 31U 22U 13

The Eq. (22) is used to find the values of critical lateral and hydrostatic pressures of FG conical shells in the framework of the STD.

a) Let the FG truncated conical shell subjected to the uniform lateral pressure, i.e. P1  0 and P2  PL . In this case, from Eq. (22), we obtain the following expression for dimensional

critical lateral pressure in the framework of the SDT:

SDT PTCLcr 

U 411  U 43  3  U 44  4  2U P2

(24)

The expression for dimensionless critical lateral pressure for a FG truncated conical shell in the framework of SDT is obtained:

SDT P1TCLcr 

U 411  U 43  3  U 44  4  2U P2 Ec

(25)

b) Let the FG truncated conical shell subjected to the uniform hydrostatic pressure, i.e. P1  P2  PH . In this case, from Eq. (22), the following expression for dimensional critical

hydrostatic pressure in the framework of SDT is obtained:

SDT PTCHcr 

U 411  U 43  3  U 44  4  2U PH

(26)

where the following definition applies:

U PH  U P1  U P2  

2m

2 1

 



 22  2  1  4n12 m12 1  e x0 ( 2 1) tan  2S2 (1  2 ) (1  2 ) 2  4m12





(27)

The expression for dimensionless critical hydrostatic pressure for the FG conical shell in the framework of SDT is obtained:

SDT P1TCHcr 

U 411  U 43  3  U 44  4  2U PH Ec

(28)

When the shear stresses are not considered in the expressions (25) and (28), we can obtain the expressions for FG truncated conical shells in the framework of the CST, and P1CST TCLcr and SDT SDT P1CST TCHcr are used instead of P1TCLcr and P1TCHcr , respectively.

The FG truncated conical shell is transformed into the FG the cylindrical shell, as

  0 , S1sin  R , m1sin  R / L1 , x0 S1  S1lnS 2 / S1  S1ln( 1  L / S1 )  L1

(29)

Substituting (29) into expressions (25) and (28), after some manipulations, we obtain the expressions for dimensionless critical lateral and hydrostatic pressures of FG cylindrical shells. Here R is radius and L1 is length of a cylindrical shell. The FG truncated conical shell is transformed into the FG complete conical shell, as

R1  10 5  0, x0  , m1  0, e ax0  0, a  0 . As R1  0 is substituted in Eqs. (25) and (28), the expressions for dimensionless critical lateral and hydrostatic pressures of FG SDT SDT complete conical shells are obtained, and P1SDT CCLcr and P1CCHcr are used instead of P1TCLcr and SDT P1TCHcr , respectively.

The values of dimensionless critical lateral and hydrostatic pressures of FG truncated and complete conical shells in the framework of the STD are found by minimizing Eqs. (25) and (28) with respect to m, n and λ. The values of the dimensionless critical lateral and hydrostatic

pressures for freely supported FG truncated and complete conical shells in the framework of the STD are found approximately at   2.4 and   4 , respectively. The minimum values of critical external pressures of freely supported cylindrical shells are obtained at λ=0 [20].

5. Numerical results

In this section, various numerical examples are presented and discussed to verify the accuracy of the present study in predicting the dimensionless critical hydrostatic and lateral pressures for FG conical shells in the framework of the CST and SDT using Maple 14 software.

5.1. Convergence study

In this subsection, the accuracy and effectiveness of the present method for the buckling analysis of fully metal and FG cylindrical and conical shells subjected to lateral or hydrostatic pressures in the framework of the CST and STD were examined with the studies of Baruch et al. [4], Shen [33] and Khazaeinejad et al. [36]. In first example, the values of dimensionless critical hydrostatic pressure for truncated conical and cylindrical shells in the framework of CST are compared with the results of Baruch et al. [4] and are tabulated in Table 1. In the computations are used the expression (28) taking into account k  , following

material

properties

Em  2.1 105 (MPa ),  m  0.3

E FG  Ec  Em ,  FG   c   m . In the comparisons, the and and

geometric

shell

characteristics

R1 / h  100; L / R1  0.5, 1.0, 2.0 .

are

The

used: material

constants and shell parameters adopted as Baruch et al. [4]. As   0 , the truncated conical shell is transformed to the cylindrical shell. It is seen from Table 1 that the values of the

dimensionless critical hydrostatic pressure which is obtained in the present study are in good agreement with the results of Baruch et al. [4]. In Table 2, the values of critical lateral pressure of metal cylindrical shell in the framework of the CST are compared with those obtained by Shen [33]. In the computations are used the expression (24) taking into account k  ,

  0 , E FG  Ec  Em ,

 FG   c   m . The material properties are taken to be Em  2  105 (MPa) and  m  0.3 , while the shell characteristics are given in Table 2. The computations data were taken from the study of Shen [33]. It can be seen from Table 2 that the values of the critical lateral pressure of present study in accordance with the results of Shen [33]. In third example, the values of critical lateral and hydrostatic pressures (in MPa) for FG (Aluminum/Alumina) cylindrical shell in the framework of the STD are compared with the results of Khazaeinejad et al. [36] and shown in Table 3. In the computations are used the expressions (24) and (26) taking into account   0  . The stresses shape functions are: f1    f 2    hsinhz / h  zcosh1 / 2 . The material properties for the alumina and

aluminum are used as ceramic and metal materials of the top and bottom surfaces of the FG cylindrical shell, respectively. The detailed material properties of alumina and aluminum are considered as Em  7  10 4 (MPa ) and Ec  3.8  105 (MPa ) , and Poisson’s ratio is assumed to be 0.3. The number in parentheses indicates the circumferential wave number (n cr). The geometric parameters and volume fraction index are shown in Table 3. These values were taken from the study of Khazaeinejad et al. [36]. Table 3 shows that there are good agreements between the present results in the framework of the SDT and the results of Khazaeinejad et al. [36].

The convergence study shows the superior performance of a closed-form solution. To further validate the accuracy and reliability of current closed-form solution, detailed numerical examples are presented in the next subsection.

5.2. Buckling analysis of FG conical shells under external pressures in the framework of SDT

In this subsection, the buckling analysis of FG truncated and complete conical shells subjected to uniform lateral and hydrostatic pressures in the framework of the CST and STD have been carried out. For the given values of the volume fraction index k , thickness-toradius ratio and semi-vertex angle  , the values of critical lateral and hydrostatic pressures are obtained. In the computations used Maple 14 software program. The mixture of materials for FG conical shells is considered to be silicon nitride (Si3N4) and stainless steel (SUS304), referred to as Si3N4/SUS304. The Young’s moduli and Poisson’s ratio of FG conical shell can be expressed as

E c  3.4843  1011 (1  3.07  10 -4 T  2.16  10 -7 T 2  8.946  10 -11T 3 )  3.22271  1011 ( Pa ) E m  2.0104  1011 (1  3.079  10 -4 T  6.5346  10 -7 T )  2.07788  1011 ( Pa )

 c  0.24  m  0.3262  (1  2.002  10 -4 T  3.797  10 -7 T )  0.317756

wherein the material properties of FGMs be independent of temperature, that is at a fixed temperature T = 300 (K), and (K), expressed in degrees Kelvin, and are unique to the constituent materials. Th material constants were taken from the studies of Reddy and Chin [14] and Shen [17].

The critical external pressures for FG truncated or complete conical shells are compared with those of metal (or ceramic) truncated and complete conical shells in the framework of the STD as well as CST by estimating the percentage differences of values of dimensionless critical external pressures, respectively, as

CST  P1SDT P1crFG  P1crH cr  P1cr  ,  P1crH P1CST cr 

   100% .  

The effect of the R2 / h on the values of dimensionless critical lateral and hydrostatic pressures of metal-rich, ceramic-rich and functionally graded truncated and complete conical shells in the framework of CST and SDT for different volume fraction index, k , are plotted in Figs. 2 and 3. The volume fraction index are taken to be k  0.5, 0.75; 1, 2, 5 and conical shells characteristics are taken to be   30  and R2 / L  5 . In addition, R1  R 2  L sin  is used for truncated conical shells. The values of dimensionless critical lateral and hydrostatic pressures of metal-rich, ceramic-rich and FG truncated and complete conical shells in the framework of the CST and SDT decrease with the increasing of the ratio, R2 / h . The influence of shear stresses on the values of dimensionless critical lateral pressure of FG truncated conical shells decreases from 31.20% to 4.73%, from 31.64% to 4.70% , from 32.13% to 4.81%, from 33.83% to 5.2%, from 35.56% to 5.63% , while this influence on the values of dimensionless critical hydrostatic pressure decreases from 16.7% to 1.79%, from 17.31% to 1.89%, from 18.49% to 1.97%, from 19.75% to 2.76% , as the ratio R2 / h increases from 20 to 50 for k  0.5; 0.75;1; 2; 5 respectively. The influences of shear stresses on the values of dimensionless critical hydrostatic and lateral pressures of FG complete conical shells almost remain constant as R2 / h increases for fixed volume fraction index k , while this influence increases from 15% to 18%, as k

increases from 0.5 to 2, then decreases for fixed ratio R2 / h . The influence of shear stresses SDT SDT on the P1TCLcr is significantly higher (almost 50%) than the P1TCHcr for fixed R2 / h and

volume fraction index k (Fig. 3). As the comparing the values of critical lateral pressure of FG truncated conical shells with those of the metal-rich truncated conical shell in the framework of the STD, the influence of SDT variation of volume fraction index, k , on the values of P1TCLcr decreases, as k  1 , while this

influence increases, as k  1 for fixed R2 / h . The influence of variation of volume fraction SDT index, k , on the values of P1TCHcr is importantly and almost remains constant, as the ratio, SDT changes, whereas, R2 / h , increases from 20 to 50. The effect of FG profiles on the P1TCLcr

this influence on the P1CST TCLcr remains constant, as the ratio, R2 / h , increases. The effects of FG profiles on the

SDT and P1CST P1TCHcr THLcr remain constant as the ratio, R2 / h , increases. The

SDT CST CST effects of FG profiles on the P1TCLcr and P1SDT TCHcr higher than the P1TCLcr and P1THLcr ,

respectively, as k  1 for fixed R2 / h (see, Fig. 2). Taking into account the effects of shear stresses, the effects of FG profiles on the critical lateral and hydrostatic pressures for FG truncated conical shells increase, while, these effects for FG complete conical shells reduce (Figs. 2 and 3). The effect of the semi-vertex angle,  , on the values of dimensionless critical lateral and hydrostatic pressures of metal-rich, ceramic-rich and functionally graded truncated or complete conical shells in the framework of the CST and SDT for different volume fraction index, k , are plotted in Figs. 4 and 5. In Fig. 4,   0  corresponds to the cylindrical shells. The volume fraction index are k  0.5, 0.75;1, 2, 5 and conical shells characteristics are taken to be R2 / h  25 and R2 / L  5 , R1  R2  L sin  .

The influences of shear stresses on the values of dimensionless critical lateral pressure of FG truncated conical shells increase from 17.26% to 24.13%, from 17.49% to 24.52%, from 17.79% to 24.98%, from 18.87% to 26.49%, from 20.03% to 28.11% for k  0.5; 0.75; 1; 2; 5, respectively, while these influences on the values of dimensionless critical hydrostatic pressure almost remain constant 10%, 10%, 10%, 11% , 12%, for k  0.5; 0.75; 1; 2; 5, respectively , as the semi-vertex angle,  , increases from 0  to 60  . The influences of shear stresses on the values of dimensionless critical hydrostatic and lateral pressures of FG complete conical shells almost same and remain constant, as the semivertex angle,  , increases for fixed volume fraction index k , while this influence increases, as the volume fraction index, k , increases from 0.5 to 2, then decreases for the fixed semivertex angle,  . As comparing the values of critical lateral and hydrostatic pressures of FG truncated conical shells with those of ceramic-rich truncated conical shells in the framework of the CST and SDT, the influences of FG profiles on the values of P1CST TCLcr are independent the variation of the semi-vertex angle,  , for all volume fraction index, k , while the influences of FG SDT profiles on the values of P1TCLcr remain constant, as k  1 and increase slightly for k  1 , as

the semi-vertex angle,  , increases. The difference between the influences of FG profiles on SDT the P1CST TCLcr and P1TCLcr is very small.

As comparing the values of critical lateral and hydrostatic pressures of FG complete conical shells with those of ceramic-rich truncated conical shells in the framework of the CST and SDT, the influences of FG profiles on the values of P1CST TCLcr are independent the variation of the semi-vertex angle,  , for all volume fraction index k , while the influences of FG SDT profiles on the values of P1TCLcr increase significantly for all volume fraction index k , as the

semi-vertex angle,  , increases from 15o to 60o by step 15o.

The values of critical lateral and hydrostatic pressures of FG truncated conical shells are compared with those of FG complete conical shells based on the SDT and CST in Fig. 6 for different semi-vertex angle,  . The volume fraction index is k  2 and conical shells characteristics are taken to be R2 / h  25 and R2 / L  5 , R1  R2  L sin  . The values of dimensionless critical lateral and hydrostatic pressures of metal-rich, ceramic-rich and FG truncated conical shells in the framework of the CST and SDT decrease with the increasing of the semi-vertex angle,  , while for complete conical shells increase, as 15     45  and SDT decrease, as   45 . The influence of shear stresses on the P1TCLcr is significantly higher than SDT the P1TCHcr for fixed semi-vertex angle,  . Furthermore, the effects of compositional profiles SDT CST CST on the P1SDT CCLcr and P1CCHcr importantly lower than the P1TCLcr and P1THLcr , respectively.

The effects of the volume fraction index, k , on the critical lateral pressures of FG truncated and complete conical shells based on the SDT and CST are plotted in Figs. 7a and b. It is noted that the outer surface of the FGM conical shells is metal-rich, and the inner surface is ceramic-rich, a fully ceramic shell ( k  0 ) has the largest critical external pressures and that the critical lateral pressures decrease, as the volume fraction index, k , increases. The material property variations are very slow for k  5 , and when k   , the conical shells is fully metal (SUS304). It is observed from Fig. 7a and 7b that the critical external pressures are quite sensitive to the change of volume fraction index, k . It is found also that the effect of SDT shear stresses on the P1TCLcr is higher than the P1SDT CCLcr , as the volume fraction index, k

increases from 0.5 to 1 and from 2 to 11.

6. Conclusion

This article presents a method to study the buckling of freely-supported FG conical shells under external pressures based on the SDT. The basic relations, the modified Donnell type buckling and compatibility equations have been obtained on the basis of SDT. The material properties of truncated conical shells are functionally graded in the thickness direction according to a volume fraction power law distribution. To solve this problem is used an unknown parameter λ in the approximation functions. One of the innovations is to achieve closed-form solutions for the critical lateral and hydrostatic pressures of freely-supported FG truncated conical shells on the basis of the SDT. The parameter λ which is included in the obtained expressions is get from the minimum conditions of critical external pressures. The different generalized values are determined for the parameter λ for critical external pressures of FG cylindrical and conical shells. Finally, the influences of shear stresses, FG profiles and conical shell characteristics on the critical external pressures are investigated.

Acknowledgment

The author thanks to Scientific and Technical Research Council of Turkey for the support of the project 113M399.

Appendix A

The operators Lij (i, j  1,2,...,4) which are given in Eq.(17) are defined as

L11 

t 11 S 24 e 4 x

L12  L13  L14 

t 18 S 24 e 4 x t 115 S 23 e 3 x t 121

t 13 t 15  2 t 16  2 t 12 t 14  3 4 4 3      x 4 S 24 e 4 x x 2  2 S 24 e 4 x x  2 S 24 e 4 x x 3 S 24 e 4 x x 2 S 24 e 4 x   2 t 19 t 110 t 113  2 t 111  3 t 112  2 4 4 3      x 4 S 24 e 4 x x 2  2 S 24 e 4 x x  2 S 24 e 4 x x 3 S 24 e 4 x x 2 S 24 e 4 x  2 t 116  3 t 117  2 t 118  t 119  t 120  2 3      x  2 S 23 e 3 x x 3 S 23 e 3 x x 2 S 23 e 3 x x S 2 e x x S 23 e 3 x  2

3

S 23 e 3 x x 2 

L21  L22  L23  L124 

t 21 S 23 e 3 x t 24 S 23 e 3 x t 27 S 23 e 3 x t 29



t 122 S 23 e 3 x

t 2   31233 x x S 2 e 

t 23 t 22 4 4 3     4 S 23 e 3 x x 2  2 S 23 e 3 x x 2 t 25  4 t 26 4 3   x 2  2 S 23 e 3 x  4 S 23 e 3 x x 2 t 28  2 3  x 2 S 22 e 2 x  2

3

S 22 e 2 x x 2 



t 210 S 22 e 2 x

t 2 3   22112 x  t 212 3 x S 2 e  

L31 

t 31 S 24 e 4 x 

L32 

t 32 t 33 t 34  2 4 4 3     4 S 24 e 4 x x 2  2 S 24 e 4 x x 2 S 24 e 4 x  2

t 35 S 24 e 4 x

t 39 S 24 e 4 x

t 36  3 t 37  2 4   x 4 S 24 e 4 x x 3 S 24 e 4 x x 2

t 310 t 311  3 t 312  2 t 313  4 4 4      4 S 24 e 4 x x 2  2 S 24 e 4 x x 2 S 24 e 4 x  2 S 24 e 4 x x 4

t 314  3 t 315  2 t 316  2 t   4 4 x 3  4 4 x 2  3 3 x 2  33173 x S 2 e x S 2 e x S 2 e x S 2 e x L33  L34 

t 319 S 23 e 3 x t 323

t 320  3 t 321  2 t 3     33223 x 2 3 3x 3 3 3x 2 x S 2 e x S 2 e x S 2 e x 3

S 23 e 3 x  3



t 324

3

S 23 e 3 x x 2 



t 325 S 23 e 3 x

t 44   2      3 3x  2 x  S 2 e  x 0.5 P1 tan    2   P tan       2 x x 2  x  S2e S2e  x t t   41 x  42 x  S 2 e x S 2 e

t 2   33263 x x S 2 e 

L41  L42 L43

L44 

 2       2  x   

t 43  S 2 e x 

where x  ln( S / S 2 ) and the following definitions apply:

(A1)

t11  C 2 ; t12  C1  C 4 ; t13  C 2  4(C1  C 4 ); t14  5C 2 ; t15  6C 2 ; t16  3(C 2  C1  C 4 ) ; t17  0; t18  C 3 ; t19  (C 3  C 6 ); t110  4C 4  4C 6  C 3 ; t111  5C 3 ; t112  6C 3 ; t113  3(C 4  C 3  C 6 ); t114  0; t115  C11; t116  C 7 ; t117  ( 2C 7  C 9 ); t118  2C 9 ; t119   I 3 ; t120  C11; t121  C12  C 8 ; t122  (C10  2C 8  2C12 ); t123  2C10 ; t 21  C 2 ; t 22  C1  C 4 ; t 23  C 4  C1  C 2 ; t 24  (C 6  C 4 ); t 25  C 3 ; t 26  C 6  C 4  C 3 ; t 27  C 9  C11; t 28  C11; t 29  C12 ; t 210  C12 ; t 211  C10 ;

(A2)

t 212   I 4 ; t 31  B1; t 32  B5  2 B2 ; t 33  ( 2 B5  4 B2 ); t 34  B5  2 B2  2 B1; t 35  B1; t 36  4 B1; t 37  4 B1; t 38  0; t 39   B4 ; t 310  B6  2 B3 ; t 311  4 B3  2 B6 ; t 312  B6  2 B3  2 B4 ; t 313   B4 ; t 314  4 B4 ; t 315  4 B4 ; t 316  cot γ, t 317   cot γ, t 318  0; t 319  B11  B7 ; t 320  B9 ; t 321   B9  B7 ; t 322  B7 ; t 323  B8 ; t 324  B12  B10 ; t 325  (B10  B8 ); t 326  B8 ; t 41  I 3 ; t 42  I 3 ; t 43  I 4 ; t 44  cot γ

in which C1  11 B1  12 B 2 , C 2  11 B 2  12 B1 , C 3  11 B3  12 B 4  21 , C 4  11 B 4  12 B3  22 , C 5  16 B5 , C 6  16 B6  226 , C 7  11 B7  12 B9  17 , C 8  11 B8  12 B10  18 , C 9  12 B7  11 B9  19 , C10  12 B8  11 B10  110 , 01 0 0 1  11 01 (A3) , B 2   2 , B3  2 2 ,    216 09 02  015 01 010 02  08 01 02 11  12 01 1 B4  , B5  0 , B 6   0 , B 7  , B8  ,    6 6

C11  111  16 B11 , C12  112  16 B12 , B1 

B9 

    

 7 02  09 01 0 0  010 01 0 012 , B10  7 2 , B11  11 , B  ,   01 12 0 0   6 6

I 3   cosh Z   cosh1 / 2dz , I 4   cosh Z   cosh1 / 2dz h/2

h/2

h / 2

h / 2

where k1j ( j  1,2,6) and kj2 ( j  7,2,...12) are defined by

2

0 2 2

h/2

k11   B11 Z z k1 dz; h / 2 h/2

h/2

h/2

k21   B12 Z z k1 dz , k61   B66 Z z k1 dz k1  0, 1, 2; h / 2

h / 2

h/2

h/2

z k2 I 1 B11 Z dz , k82  

k72  

h / 2

h / 2

h/2

h/2

z k2 I 2 B11 Z dz , k112  

k102  

h / 2

h / 2

z k2 I 2 B12 Z dz , k92  

z k2 I 1 B12 Z dz ,

h / 2

h/2

z k2 I 1 B66 Z dz , k122  

h / 2

(A4)

z k2 I 2 B66 Z dz,

k 2  0,1,

Appendix B

The parameters U ij ( i , j  1,2,...,4 ) , U P1 and U P2 are given in Eq. (20) and defined by

 



 

    2   m ,

U 11  2m12 t11 3  1  13  2m12   4   1  m14  t12 n12  2    2  m12







 m12 3t13  2t16 n12  t14   12 4  5   12 4  7   2t15

U 12  



  m   1 1  e

m12 2  12  m12 2  1 1  e 2 x0 (1 )



4 S 2   1



2

2 1



x0 (1 2  )

t 3  4 18

3

2

2 1

 2   2 m12  m14







 



 t19 n12    2   m12  t110 n12  t111 23  2m12  3 2  m12  t112    2   m12  t113 n12 ,

  m  



   2  2m  t 2  1  2m , 2  1 2  1  m  1  e t  1  2   2m S

U 13  m1 t115 2  1  2m12 n12  t116 2  13  3m12  2m14  t117

2 1

2

3

2 1

2 1

118

2 x0 (1 2  ) 1 1 2m1 2  12  12  4m12 1  e x0 (1 2 ) 2













2 1

119

U 14  n1  2t121 m12   1  m12  t122 m12  2t123 m12



2 2



 t120 2  1n12 ,

U 21



U 22  

U 23 

U 24





m12 n12 1  e 2 x0  t n 2  t 22 2  1  m12  t 23 2 2 2 21 1 4   m1 S 2







m12 n12 2  m12

2  1

2



m

t m1n12  27 

 2 1

4m12



2  1S 



1  e x0 12  1 e

2

 2 x0





2t (  1)  m  2t 2 1

24

 2 2  1  2m12 1  e x0 12 

2  1



 4m12 2  1S 2

2

 

1  e 2 x0









2 25 n1

U 31 













  



1 m12 t 31n14  t 32 n12 2  1  m12  t 33 n12  t 34 n12  t 37 m12  2  1 4 



 



t 28   4 

m12 n1  t 29 m12  2  t 211n12 t 212 S 22 m12  2 1  e 2 1x0    4     1  12  m12 1  e 2x0



 t 26

 t 35 m14    1 3  1  2  3  1m12  t 36 2m12   1  32  3m12  23 U 32  

3



m12 m12  2

2  1 2  1

 t 311n12  t 314

U 33

4

3



4m12



S

1  e x0 (12 ) 1 e

2

 2 x0

2t



 2t 310 2    m12

4 39 n1

2t 312 n12

 2t 313 2  3 3  2m12    1  m14

 3 

4m12 



2





m12

 2t 



2

315

 

m12





 t m 4  m 316

2 1

2

2 1

,





m1 m12  2  t 319 n12  t 320 2  m12  t 321  t 322 , 4

U 34 

U 41



2











m12 t 326 n1  t 323 n13  t 324 n1 m12  2 , 4





t 44 m12 4m12  (2  1) 2 (2  1) 1  e 2x0  , 4S 2  1  e  x0 ( 2 1)





U P1  0.5m12 (2m12  22  2  1) tan  , U P2   2n12  1 m12 tan  ,

 

U 43  m1 t 41 2  1  U 44  2t 43 m12 n1

2m12

 t

42

2  1

(B1)

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Table and Figure Captions

Table 1 Comparison of the values of dimensionless critical hydrostatic pressure of fully metal cylindrical and conical shells in the framework of the CST ( R1 / h  100) . Table 2 Comparison of the values of critical lateral pressure of fully metal cylindrical shells in the framework of the CST Table 3 SDT SDT Comparison of the values of Pcrcyl and PLcrcyl (in MPa) for different R / h and power law

index k with L1 / R  5 . Fig. 1. Geometry of FG truncated conical shell and coordinate system. Fig. 2. The effect of R2 / h on the critical (a) lateral and (b) hydrostatic pressures of FG truncated conical shells with different volume fraction index based on SDT and CST. Fig. 3. The effect of R2 / h on the critical (a) lateral and (b) hydrostatic pressures of FG complete conical shells with different volume fraction index based on SDT and CST.

Fig. 4. The effect of the semi-vertex angle,  , on the critical (a) lateral and (b) hydrostatic pressures of FG truncated conical shells with different volume fraction index based on SDT and CST. Fig. 5. The effect of the semi-vertex angle,  , on the critical (a) lateral and (b) hydrostatic pressures of FG complete conical shells with different volume fraction index based on SDT and CST. Fig. 6. The comparison the values of critical lateral and hydrostatic pressures of FG truncated and complete conical shells based on SDT and CST. Fig. 7. The variation of the critical lateral and hydrostatic pressures of FG truncated and complete conical shells based on SDT and CST for (a) 0  k  1 and (b) k  2 .

Table 1

6 P1CST TCHcr  10 , ( ncr )

L / R1  0.5

L / R1  1

L / R1  2

Baruch et

Present

Baruch et

Present

Baruch et

Present

al. [4]

study

al. [4]

study

al. [4]

study

Cylinder 0o

21.06(11)

20.47(11)

9.838 (8)

9.588(8)

4.744(6)

4.618(6)

10o

19.40(11)

19.392(11)

8.569(9)

8.5559(9)

3.740(7)

3.699 (7)

30o

14.55(11)

14.506(11)

5.843(9)

5.8236(9)

2.237(8)

2.147(8)

50o

8.813(11)

8.7244(11)

3.285(9)

3.2548 (9)

1.164(8)

1.1366 (9)



Table 2

Shen [33]

L/ R R/h

Present study

CST PLcrcyl  10 4 (MPa), ( ncr )

300 1272.597(11) 1273.351(11) 1 3000

3.8144(20)

3.814807(20)

300

402.6016(7)

412.4069(7)

3000

1.2511(12)

1.255621(12)

300

239.0987(5) 239.28662 (5)

3

5 3000

0.7482(9)

0.7479701(9)

Table 3

Khazaeinejad et al. [36]

k

0

0.5

1

5

Present study

R/h 

10

50

100

10

50

100

SDT PLcrcyl

242.946(2)

4.219(3)

0.722(4)

242.612(2)

4.217(3)

0.7215(4)

SDT PHcrcyl

231.521(2)

4.128(3)

0.713(4)

231.193(2)

4.126(3)

0.7127(4)

SDT PLcrcyl

163.73(2)

2.854(3)

0.479(4)

163.557(2)

2.853(3)

0.4787(4)

SDT PHcrcyl

156.03(2)

2.793(3)

0.473(4)

155.859(2)

2.792(3)

0.4729(4)

SDT PLcrcyl

128.411(2)

2.243(3)

0.373(4)

128.255(2)

2.242(3)

0.3726(4)

SDT PHcrcyl

122.372(2)

2.195(3)

0.368(4)

122.219(2)

2.194(3)

0.368(4)

SDT PLcrcyl

79.224(2)

1.375(3)

0.236(4)

78.919(2)

1.374(3)

0.236(4)

SDT PHcrcyl

75.498(2)

1.345(3)

0.233(4)

75.204(2)

1.344(3)

0.233(4)

Figure 1

Figure 2

10

SDT

CST

Metal k=0.5 k=0.75 k=1 k=2 k=5 Ceramic

8

6

Metal k=0.5 k=0.75 k=1 k=2 k=5 Ceramic

4

2

0 20

30

40

50

(a)

6

SDT

CST

Metal k=0.5 k=0.75 k=1 k=2 k=5 Ceramic

4

Metal k=0.5 k=0.75 k=1 k=2 k=5 Ceramic

2

0 20

30

40

(b)

50

Figure 3

0.8

SDT

0.7

CST

Metal k=0.5 k=0.75 k=1 k=2 k=5 Ceramic

0.6 0.5

Metal k=0.5 k=0.75 k=1 k=2 k=5 Ceramic

0.4 0.3 0.2 0.1 0 20

30

40

50

(a)

0.7

SDT 0.6

CST

Metal k=0.5 k=0.75 k=1 k=2 k=5 Ceramic

0.5 0.4

Metal k=0.5 k=0.75 k=1 k=2 k=5 Ceramic

0.3 0.2 0.1 0 20

30

40

(b)

50

Figure 4

8

SDT

CST

Metal k=0.5 k=0.75 k=1 k=2 k=5 Ceramic

7 6

Metal k=0.5 k=0.75 k=1 k=2 k=5 Ceramic

5 4 3 2 1 0

15

30

45

60

γ(angle)

(a)

4.0

SDT

CST

Metal k=0.5 k=0.75 k=1 k=2 k=5 Ceramic

3.5 3.0

Metal k=0.5 k=0.75 k=1 k=2 k=5 Ceramic

2.5 2.0 1.5 1.0 0.5 0

15

30

γ(angle)

(b)

45

60

Figure 5

0.8

SDT 0.7

CST

Metal k=0.5 k=0.75 k=1 k=2 k=5 Ceramic

0.6 0.5

Metal k=0.5 k=0.75 k=1 k=2 k=5 Ceramic

0.4 0.3 0.2 0.1 15

30

45

60

γ(angle)

(a)

0.7

SDT

CST

Metal k=0.5 k=0.75 k=1 k=2 k=5 Ceramic

0.6

0.5

Metal k=0.5 k=0.75 k=1 k=2 k=5 Ceramic

0.4

0.3

0.2

0.1 15

30

45

γ(angle)

(b)

60

Figure 6

k=2

5

SDT PL TC PH TC PL CC PH CC

4

CST PL TC PH TC PL CC PH CC

3

2

1

0 15

30

45

γ(angle)

60

Figure 7

6 5

SDT

CST

TC

TC

CC

CC

4 3 2 1 0 0.5

0.75

1

k

(a)

SDT

5

4

CST

TC

TC

CC

CC

3

2

1

0 2

5

8

k

(b)

11