Lat. Am. j. solids struct. vol.14 número4

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Abstract

This paper deals with free vibration analysis of thick cylindrical composite sandwich panels with simply supported boundary condi-tions based on a new improved higher-order sandwich panel theory. The formulation used the third-order polynomial description for the displacement fields of thick composite face sheets and for the dis-placement fields in the core layer based on the disdis-placement field of Frostig's second model. In this case, the unknowns were coefficients of the polynomials in addition to displacements of the top and bot-tom face sheets. The fully dynamic effects of the core layer and face sheets were also considered in this study. Using Hamilton's principle, the governing equations were derived. Moreover, the effect of some important parameters such as those of thickness ratio of the core to panel, the length to radius ratio of the core and composite lay-up sequences were investigated on free vibration response of the panel. The results were validated by those published in the literature and with the finite element results obtained by ABAQUS. It was shown that thicker panels with thicker cores provided greater resistance to resonant vibrations. Moreover, the effect of increasing face sheets’ thicknesses in general was the significant increase in fundamental natural frequency values.

Keywords

Cylindrical sandwich panel, Free vibration, Improved higher-order theory, Flexible core, Analytical analysis.

Vibration Analysis of a Cylindrical Sandwich Panel with Flexible

Core Using an Improved Higher-Order Theory

1 INTRODUCTION

The use of sandwich structures has increased in recent years in aerospace, naval, civil, transportation, and other industries which require stiff and light-weight structural ingredients. Sandwich structures are constructed of three layers. They are usually composed of two metallic or composite laminated materials: face sheets and a foam core or a low-strength honeycomb core. The materials with a high

A.R. Pourmoayed 1 K. Malekzadeh Fard * M. Shahravi 2

Department of Structural Analysis and Simulation, Aerospace Research Institute, Malek Ashtar University of Technology, Post Box: 13445-768, Tehran, IRAN.

email: 1 pourmoayed @mut.ac.ir *_{[email protected]} 2_{[email protected]}

http://dx.doi.org/10.1590/1679-78253410 Received 02.10.2016

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strength are usually used for face sheets, whereas the core layer is made of a low-specific-weight mate-rial which may be much less stiff or strong than the face sheets (Librescu and Hause, 2000).

NOMENCLATURE

, ,

x qz _{Longitudinal, circumferential and radial coordinates}

( , , )

i= c t b _{Indices for core, outer (top) and inner (bottom) face} _sheets

, ,

i i i

u v w _{Displacements in longitudinal, circumferential and radial directions}

0i, 0i, 0i

u v w Mid-plane displacements in longitudinal, circumferential and transverse

di-rections

,

i i i

xx  zz



Strains in face sheets

0

,

0

i i i

xx  zz



Mid-plane strains in face sheets and the core

,

i i i

xq xz qz



Shear strains in face sheets and the core

ij

Q Reduced stiffnesses referring to the principal material coordinates

ij

Q Transformed reduced stiffnesses

j ii



Normal and transverse stresses in the face sheets, i=( , , ),x qz j=( , )t b

c ii



Normal and transverse stresses in the core, i=( , , )x qz

j j j x, xz, z

   Shear stresses in the face sheets, j =( , )t b

,

c c c

x xz z



Shear stresses in the core

,

,...

i i i i

xx xx xx xx

N

M

P H

Stress resultants in the face sheets, i=( , )t b

,

,...

c c c c

xx xx xx xx

N

M

P H

Stress resultants in the core

l jk

Q

The reduced stiffness coefficients of the lth composite layer of each face sheet

K Kinetic energy

U Strain energy

i

l _{the number of composite layers in each face sheet}

ji

I Moment of inertia

i

 Mass density

, ,

t c b

dV dV dV Volume element of the top face sheet, the core and the bottom face sheet, respectively

L Length of cylindrical sandwich panel

h Total thickness of cylindrical sandwich panel

t

h _{Thickness of the top face sheet}

b

h Thickness of the bottom face sheet

(3)

K

 Stiffness matrix



Natural frequency

In recent years, to describe the dynamic and static behavior of these structures, various theoret-ical models have been developed. Understanding the free vibration behavior of structures is essential for preventing the occurrence of resonance, and for optimal designing. Moreover it is important to Recovering new types of materials for the face sheets and core in order to reduce failure modes and obtain optimum weight sandwich structural. Different approaches may be used for modeling sand-wich panels. The first approach is equivalent single layer (ESL) models and classical models (see Reddy, 1997; Mindlin, 1951; and Wang, 1996). These theories often obtain inaccurate results when used for the analysis of sandwich panels with flexible cores that are soft in the vertical direction (Frostig and Thomsen, 2008). The second approach is the first-order shear deformation theories (FSDT). When the face sheets in the sandwich panel are thin, the FSDT model obtains good results for the analysis of sandwich panels with flexible cores (see Malekzadeh et al., 2015). Reissner (1985), Noor and Burton (1989), Reddy (1990), and Kant and Swaminathan (2001) have reviewed these developments. There are also other theories studied by some researchers. Biglari and Jafari (2010) studied the free vibration of doubly curved composite sandwich panels with soft cores using the mixed theory. Singh (1999) applied the Rayleigh–Ritz method for free vibration analysis of doubly curved open deep sandwich shells. Meunier and Shenoi (2001), Nayak et al. (2002) and Carrera (2004) used a ‘‘zig-zag’’ displacement pattern for the modeling of layered plates and shells. Garg et al. (2006) investigated the free vibration analysis of simply supported composite and sandwich dou-bly curved shells. Their formulation included Sander's theory based on an equivalent single-layer approach. In order to include the three-dimensional fully dynamic modeling of the flexible thick core of the panels, researchers have usually used a High-order Sandwich Panel Theory (HSAPT). In this context, researchers such as Frostig and Thomsen (2004), Bozhevolnaya and Frostig (2001), and Rabinovitch et al. (2003) have used HSAPT in various structural problems. Malekzadeh et al. (2014) studied the improved high-order bending analysis of doubly curved sandwich panels subject-ed to multiple loading conditions. Rahmani et al. (2010) applisubject-ed a higher-order sandwich panel the-ory in order to study the free vibration analysis of an open single curved composite sandwich panel with a flexible core.

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This study investigated the free vibration analysis of doubly curved thick composite sandwich panels using a new improved higher-order sandwich panel theory based on the second computation-al model of Frostig (2004). In the present formulation, the top and bottom face sheets can be thick or thin. The in-plane stresses of the core were also considered. In this study, the analytical solution of the displacement field of the core was presented in terms of polynomials with unknown coeffi-cients according to the second computational model of Frostig (2004). Furthermore, the formulation included accurate stress-resultant equations for composite sandwich structures, in which the term (1zc/ )Rc was imported in Eq.3 and exactly integrated. These coefficients could be very important in the structural analysis of thick cylindrical sandwich panel structures.

2 THEORY AND FORMULATION

2.1 Basic Assumptions

Consider a cylindrical thick composite sandwich panel which is composed of two composite laminat-ed thick face sheets and a flexible core layer. The geometry of the panel and the coordinates are shown in Fig. 1. In this figure, indices t and b refer to the top and bottom face sheets of the sand-wich panels, respectively. Moreover, the thicknesses of the top face sheet, the bottom face sheet, the core layer, the total thickness of sandwich panel, the intermediate radius of the core, the intermedi-ate radius of the top face sheet, the intermediintermedi-ate radius of the bottom face sheet, and the length of the sandwich panel are presented by,ht,hb,hc, h,Rc,Rt,Rb and L, respectively. The displacement fields in face sheets are u, v and w in the directions of x(longitudinal),



(circumferential) and z

(radial), respectively. They are measured upward from the midplane of the face sheets (Reddy, 2003). Face sheets are thick laminated composite and orthotropic structures. The face sheets and the core are assumed to be perfectly bonded, i.e. there is no relative displacement between the face sheets and the core interfaces.

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2.2 Sandwich Kinematics

The displacements fields in face sheets, , , , , , , , and , , , in the direction of

x,



and z, respectively are explained as follows:

2 3

0 1 2 3

2 3

0 1 2 3

2

0 1 2

( , , , ) ( , , ) ( , , ) ( , , ) ( , , )

( , , , ) ( , , ) ( , , ) ( , , )

, , t : top f

i i i i

i i i

u x z t u x t z u x t z u x t z u x t

v x z t v x t z v x t z v x t z v x t

w x z t w x t z w x t z w x t i t b

q q q q q

q q q q

= + + +

= + +

= ace sheet , : bottom face sheet

( (

) )

2 2

t c b c

t b

b

h h h h

z = -z + z = +z +

(1)

where _u₀i _,

0i

v denote inplane displacement and _w₀i _{out of plane displacement, respectively, in midle}

surface. The kinematic equations for the strains in the face sheets are as follows (Bert, 1967, and Soykasap et al., 1996):

i i

, , ,

2 ,

1 2

i i i i i

xx zz

i i

i i i i

i

i i i i

xz xz

i i i i i

z z

i i i

x x

u v w

x z

v u

x

w u

x z

w v

z w

R R

R

v

R R

qq

q q

e e e

q

g e

q

g e

q

¶ ¶ ¶

= = + =

¶ ¶ ¶

¶ ¶ = = +

¶ ¶ ¶ ¶ = = +

¶ ¶

= = - +

¶ ¶

(2)

Based on the second Frostig’s model, the displacement field components of the core layer are derived as:

2 3

2 c 3

0 1 2 3

2

0 1

0

2 1

( , , , ) ( , , ) ( , , ) ( , , ) ( , , )

( , , , ) 1 ( , , ) ( , , ) ( , , ) ( , , )

( , , , ) ( , , ) ( , , ) ( , , )

c c c c

c c c

c

c c c c

c

c c c

x z t u x t z x t z x t z x t

v x z t v x t z x t z x t z x t

w

u u u u

z

v v v

R

w w

x z t w x t z x t z x t

q q q q q

q q q q

= + + +

æ _ö÷ ç _÷

=ç_ç + _÷_÷ + + + çè ø

= + +

(3)

In Eq.3, it is seen that the number of unknown variables is 11. Based on small deformations, the kinematic relations of the core are as follows:

æ ö

¶ _ç ¶ _÷ ¶

÷

= = _æ _öçç_ç + _÷_÷ = ¶ _ç ₊ _÷_÷è ¶ ø ¶

ç _÷ ç ÷ çè ø

1

, ,

1

c c c c c c c

xx zz

c c c

c

u v w w

x z R R z

R

qq

e e e

q

¶ ¶ ¶ ¶

= = +_æ _ö = = + ¶ _ç _÷_÷ ¶ ¶ ¶

+ ç _÷ ç ÷ çè ø

1

2 , 2

1

c c c c c c c c

x x xz xz

c c

c

v u w u

x z R x z

R

q q

g e g e

q

(6)

æ _¶ ö_÷ _¶

ç _÷

= =_æ ç_ç - _÷_÷+ öç ¶è ø ¶ ÷

ç ₊ _÷ ç _÷ ç ÷ çè ø

1 2

1

c c c c c

z z

c c c c

c

w v v

z R R z

R

q q

g e

q

2.3 Compatibility Conditions

Based on the positive upward direction, the compatibility conditions in the interface of the core and the top and bottom face sheets could be written as below:

2 2 ₂ ₂

,

t c

b c

t c

b c

t c

b c

t c

b c

t c

b c

t c

b c

h h

t z c z _b _z h _c _z h

h h

t _z c _z _b h _c h

z z

h h

t z c z _b _z h _c _z h

u u _u _u

v v _v _v

w w _w _w

-

-= = ₌ ₌

-

-= = ₌ ₌

-

-= = ₌ ₌

= ìï ₌

ìï ï

ï ï

í í

ï ï

î _ïî

= ₌

(5)

According to Eq.5, the relation between displacement dependent parameters is extracted in the core:

2 2 3 3

0 0 1 1 2 2 3 3 0

2 ₂

2 2 3 3

0 0 1 1 2 2 3 3 1

3 ₃

2 2 3

0 0 1 1 2 2 3

2

8( ) 4( ) 2( ) ( ) 16

4

8( ) 4( ) 2( ) ( ) 8

2

8( ) 4( ) 2( ) (

b t b t b t b t c

c b t b t b t

c

t b t b t b t b c

c t b t b t b c

c

t b b t b t b

c b t b t b

u u h u h u h u h u h u h u u

u

h

u u h u h u h u h u h u h u h u

u

h

v v h v h v h v h v h v

v

+ + - + + + - -=

- - + + - - + -=

+ + - + + +

-= 3 3 0

2

2 2 3 3

0 0 1 1 2 2 3 3 0 1

3 ₃

2 2

0 0 1 1 2 2

1

2 2

0 0 1 1 2 2 0

2

) 16

4

8( ) 4( ) 2( ) ( ) 8 8

2

8( ) 4( ) 2( )

8

8( ) 4( ) 2( ) 16

t c

t b t b t b t b c c c

t b t b t b c

c c

c

t b t b t b

c t b t b

c

b t b t b t c

c b t b t

h v v

h

v v h v h v h v h v h v h v v h v

R v

h

w w h w h w h w h w

w

h

w w h w h w h w h w w

w

-- - + + - - + - -=

- - + +

-=

+ + - + +

-=

2

4h_c

ìïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï íï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïï ïïïî

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According to this equation, the number of unknown variables in the core is decreased from 11 to 5. Therefore, the total number of unknowns in the core and the face sheets is reduced to 27:

0 0 0 0 0 1 1 1 1 1 1 2 2 2

t c c c

2 2 2 3 3 3 3 0 1 0 1 0 0

, , , , , , , , , , , , , , ,

, , , , , , , , , , ,

b b b t t t b b b t t t b b b

t t t b b t c c

v w u v w u v w u v w u v w

v w u v u v v

u

u u u v w

ìïïï íï

ïïî (7)

2.4 The Stress-Strain Relations and Stress Resultants

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11 12 13

12 22 23

13 23 33

44

55

66

0 0 0

0 0 0 0 0

i i i

xx xx

i i i

zz zz

i

x x

i

xz xz

i

z z

k

k _Q _Q _Q

Q Q Q

Q

qq qq

q q

s e

s g

é ù

ì ü ì

ï ï _ê _ú ï

ï ï ï

ï ï ê ú ï

ï ï _ê _ú ï

ï ï ï

ï ï _ê _ú ï

ï ï ï

ï ï ê ú ï

ï ï ₌ _ê _ú ï

í ý _ê _ú í

ï ï ï

ï ï _ê _ú ï

ï ï ï

ï ï ê ú ï

ï ï _ê _ú ï

ï ï ï

ï ï _ê _ú ï

ï ï

ï ï ê ú

î þ _ë _û î

k üïï ïï ïï ïï ïý ïï ïï ïï ï ï ï ï _ïþ

(8)

where Q_mn( ,m n =1, 2, 4) is the reduced stiffness coefficients and Q_mn( ,m n=3,5, 6) is the trans-verse shear stiffness coefficients. The values of stiffness coefficients as presented by Malekzadeh et al. (2010) and Garg et al. (2006) are as follows:

4 2 2 4

11 11 12 44 22

2 2 4 4

12 11 22 44 12

2 2

13 13 23

cos 2( 2 )sin cos sin

( 4 )sin cos (cos sin )

cos sin ,

i i i i i

i i i

Q Q Q Q Q

Q Q Q

q q q q

q q

= + + +

= + - + +

= +

4 2 2 4

22 11 12 44 22

2 2

23 13 23

21 12

31 13

32 23

33 33

2 2 2 2 2

44 11 22 12 44

2 2

55 55 66

2

66 55

sin 2( 2 ) sin cos cos

sin cos

( 2 )sin cos (cos sin )

cos sin

sin

i i i i i

i i i

i i i i i i i i

i i i i i

i i i

i i

Q Q Q Q Q

Q Q Q

Q Q

Q Q Q Q Q

Q Q Q

Q Q

q q q q

q q

q q q q

q q

q

= + + +

= +

= = = =

= + - +

-= +

= + 2

66cos

,

i

Q i t b

q

=

(9)

Stress-strain relations in the core are as follows (Khalili et al., 2012):

11 12 13

12 22 23

13 23 33

44

55

66

0 0 0

0 0 0 0 0

c c c c c

xx xx

c c c c c

zz zz

c

c c

x x

c

c c

xz xz

c z

k k

c c

z

Q Q Q

Q

qq qq

q q

s e

t g

ì ü é ù ì

ï ï ï

ï ï ê ú

ï ï _ê _ú

ï ï

ï ï _ê _ú

ï ï

ï ï ê ú

ï ï _ê _ú

ï ï

ï ï ₌ _ê _ú

í ý _ê _ú í

ï ï

ï ï _ê _ú

ï ï

ï ï ê ú

ï ï _ê _ú

ï ï

ï ï _ê _ú

ï ï

ï ï ê ú

ï ï ë û

î þ

k üï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï _ïý ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï î þ

(10)

(8)

(

)

(

)

(

)

(



)

(



)

(



)

  



1 2 3 3 2 1 2 1 3 1 2 3 1 3 1 2 1 3 2

1 1 1 2 1 3

2 1 3 3 1 2 3 2 1 2 3 1 2 1 2 2 1

2 2 2 3 3 3

4 4 1 2 5 5 1 3 6 6 2 3

1 , , 1 1 , , , , 1

j J J j J J J j J J J

j j j

J J J

j J J j J J J j J J

j j j

J J J

j j j j j j

J

E E E

Q Q Q

E E E

Q Q Q

Q G Q G Q G

J J J J J J J J

J J J J J J J

J - + + = = = - + -= = = = = =

= - _{1 2} _{2 1} _{2 3} _{3 2} _{3 1 1 3} 2 _{2 1} _{3 2} _{1 3} , ,

J J J J J J J J J

j t b c

J - J J - J J - J J J

=

(11)

Based on the relations of strain potential energy of face –sheets, the stress resultants in the face –sheets are calculated as follows:

2 2 2 2 3 3 2 2 1 1 , i i i i i i h h xx i i i i

xx i i

xx i

i i

xx h h

i i i i xx N N z z M M dz z z P P z z H H qq qq qq qq qq s s -

-ì ü ì ü ì ü ì ü ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï₌ ï ï ï ï₌ ï í ý í ý í ý í ý ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ïî þ ï ï ïî

î þ î þ

ò

ïïïdz_i ,

ïï ïï ïïþ 2 2 2 2 3 2 , 1 , 1 i i i i h i zz i zz i i i zz h i h x i xz i i

x i i

x i xz

i

x h i

xz i i x N dz z M N Q z M dz S z P R z H q q q q q s s

-ì ü ì ü ï ï ï ï ï ï

ï ï₌ ï ï í ý í ý ï ï ï ï ï ï ï ïî þ ï ï

î þ ì ü ì ü ï ï ï ï

ï ï ï ï ì_ï ü_ï ï ï ï ï _ï _ï ï ï ï ï _ï _ï ï ï ï ï _ï _ï ï ï₌ ï ï ï ï í ý í ý í ý ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï_ïî ï_þ ï ï ï ï

ï ï ï ïî þ î þ

ò

2 2 2 2 2 3 2 1 , 1 , , i i i i h i

xz i i h i i h z i i z i z i i i z h i i z z dz z Q z

S i t b

dz z R z V q q q q q s s -ì ü ï ï ï ï ï ï ï ï ï ï = í ý_{ï ï} ï ï ï ï ï ï ï ï ï î þ ì ü ì ü

ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï

ï ï ï ï = ï ï₌ ï ï

í ý í ý ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ïî þ î þ

ò

(12)

And based on the relations of strain potential energy of the core, the stress resultants in the core are calculated as follows:

-

-ì ü ì ü ì ü ì

ï ï ï ï ï ï ï

ï ï ï ï ï ï

ï ï _æ _öï ï ï ï

ï ï ï ï ï ï

ï ï₌ ç_ç ₊ ÷ï ï_÷ ï ï₌

í ý _ç _÷_÷í ý í ý í ï ï _çè _øï ï ï ï

ï ï ï ï ï ï

ï ï ï ïî þ ï ï

î þ î þ

ò

2 2

ò

2 2

3 3

2 2

1 1

1 ,

c c c c c c h h xx c c c c

xx c c c

xx c

c c

c

xx h h

c c

xx

N N

z z

M z M

dz z z R P P z z H H qq qq qq qq qq s s - -üïï ï ï ï ï ï ï ï ïý ï ï ï ï ï ï ï ï ï ï ï ï î þ

ì ü ì

ï ï ï ï ï ï ï ï

ì ü ì ü

ï ï æ öï ï ï ï æ ö

ï ï ÷ ï ï ÷

ï ï₌ ç_ç ₊ _÷ï ï ₌ ç_ç ₊ _÷ í ý _ç _÷_÷í ý í ý _ç _÷_÷í ï ï ç_è _øï ï ï ï ç_è _ø ï ï ï ïî þ ï ï

ï ï

î þ ï_ï ï_ï

ï ï ï ï î þ

ò

* * 2 2 2 * 3 * 2 2 1 , 1 1 1 , c c c c c c

h x h

c c

c

zz c c x c c

zz c x

c c

zz _h x _h

c c x dz N z

N z M z

(9)

- -ì ü ì ü

ï ï ï ï

ï ï ï ï ìï üï ì üï ï

ï ï ï ï _ï _ï _{ï ï}

ï ï ï ï _ï _ï _æ _{öï ï}

ï ï ï ï _ï _ï _{ï ï}

ï ï₌ ï ï ï ï₌ ç_ç ₊ ÷_÷ï í ý í ý í ý _ç ÷_÷í ý ï ï ï ï ï ï _çè _øï

ï ï ï ï ï_ï_î ï_ï_þ _ïï_î ï ï ï ï

ï ï ï ïî þ î þ

ò

* 2 2 * 2 * 2 3 2 2 , 1 1 1 c c c c c h h x c xz c c

x c c c c

x c xz xz c

c

c c

x h c h

xz c c c x N Q z M z

dz S z

z R P R z z H q q q q q s s - -ï ïï ïïïþ ì ü ì

ï ï ï ï ï ï

ì ü ì ü

ï ï æ öï ï ï ï ï ï ï ÷ï ï ï ï

ï ï₌ ç_ç ₊ _÷ï ï ₌ ï í ý _ç ÷_÷í ý í ý í ï ï _çè _øï ï ï ï ï

ï ï ï ï ï ï

ï ï ï ï_{ï ï} ï ï

ï ï î þ

î þ _ï_ï _ï_ï

î î þ

ò

* * 2 2 2 * 2 3 * 2 2 , , 1 1 1 c c c c c c

h z h

c

xz c

c z

c c c c

xz xz c c c z

c

c z

c h h

xz c c

c z dz Q Q z S z

S z dz

z R R R z z H q q q q q s s -üïï ïï ïï ýï ï ï ï ï ï ï ï ï ïþ

ì ü ì ü

ï ï ï ï

ï ï æ öï ï ï ï ÷ï ï ï ï₌ ç_ç ₊ _÷ï ï í ý _ç _÷_÷í ý ï ï _çè _øï ï

ï ï ï ï

ï ï ï ï_{ï ï}

ï ï î þ

î þ

ò

2 2 2 , 1 1 c c c h c z

c c c

z z c c

c c h z c dz Q z

S z dz

R R z q q q q s

3 GOVERNING EQUATIONS

The governing equations of motion for the face sheets and the core are derived using Hamilton's principle of minimization of the Lagrangian L of the deformed system:

0 0

dt dt 0

T T

L K U

d = é_ëd -d ù_û =

ò

(14)

where K is the kinetic energy, U is the strain energy and



denotes the variation operator. The variation of kinetic energy is extracted as follows:

2 i

2

, , ₂

c 2 ( ) ( ) , , i i i c c c h i i A h h i t b c

c c

A h

i i

i i i i i i

c c

i

c c c

i i i i i i

c

c c c c

dz dA

K

dz dA

dA R dx d i t b

dA R dx d dV R dx

u u v v w w

u u v v

d dz dA dz

dV w w r d r q q

d d d

d d q d -= -é ù ê ú ê ú ê ú ê ú ê ú ê ú

= - _ê _ú

ê ú

ê

+ +

+ + ú

+ ê ú ê ú ê ú ë û = = = = = =

ò

å

ò

     

∬

1 c c c c 1 c c c

c c

z z

R dx d dz dA dz

R q R

æ ö_÷ æ ö_÷ ç ₊ _÷ ₌ç ₊ _÷

ç _÷ ç _÷

ç ç

è ø è ø

(15)

(10)

(

)

(

)

, ,

,

i

c

i i i i i i i i i i i i

xx xx zz zz x x xz xz z z i

V

i t b c c c c c c c c c c c c c

xx xx zz zz x x xz xz z z c

V

i i i

c c

dV

U

dV

dA R dx d i t b

dA R

qq qq q q q q

s de s de s de t dg t dg t dg

d

s de s de s de t g t dg t dg

q

=

é ù

ê ₊ ₊ ₊ ₊ ₊ ú

ê ú

= _ê _ú

ê₊ ₊ ₊ ₊ ₊ ₊ ú

ê ú

ë û

= =

=

ò

å

ò

,

1

c

i i i

c

c c c

c

dx d dV dAdz

z

dV dA dz

R q

= æ _ö÷ ç _÷ =ç_ç + _÷_÷ çè ø

(16)

The governing equations for a cylindrical sandwich panel composed of a flexible core and thick composite face sheets are derived. Hence, after integration by parts and some algebraic manipula-tion, twenty-seven equations of motion are extracted, some of which are as follows:

At the top face sheet:

0

, , ₂ , ₃ , ₂ x, ₃ x, ₂ xz ₃ xz

t c c

t t t t 2 3 4 5

0 0 1 2 3 ₂ ₃ ₄ ₅

c

3 4 5

2 3

4

1 0

:

1 2 4 2 4 4 12

N N P P H S R

R R h R h h h

2I 4I 8I 16I

I I I I

h h h h

2I 4I 8I

h h h

t

x

t t c c c c c c

c c c c c c

c c c c

t t t t

x x x xx x xx x

c

c c c

h h

u u u u

u

u H

q q q q q q

+ + + + + -

-æ _ö÷

ç _÷

ç

= + + + +_ç + - - _÷_÷ +

ç ÷

è ø

+ -

-    



6 4 6 4 5 6

5 3 6 4 5 6

c c 2

b 4 b 6 t 4 t 5 t 6 b 4

3 6 4 5 6

c

1 0 0

1 1

16I 4I 16I 4I 16I 16I

h h h h h h

2h I 8h I 2h I 8h I 8h I h I

h h h h h h

c c c c c c

c b t

c c c c c c

c c c c

b t

c c c c

u u u

u u

æ ö_÷ æ ö_÷ æ ö_÷

ç _÷ ç _÷ ç _÷

ç ÷ +ç - ÷ +ç + + ÷

ç _÷ ç _÷ ç _÷

ç ÷ ç ÷ ç ÷

è ø è ø è ø

æ _÷ö æ ö_÷

ç _÷ ç _÷

ç ç

+_ç - _÷_÷ + -_ç - - _÷_÷ +

ç ÷ ç ÷

è ø è ø

  

  2b 6

4 6

2 2 2 3 c 3 3 3 3

t 4 t 5 t 6 b 4 b 6 t 4 t 5 t 6

4

2

2 3

5 6 4 6 4 5 6 3

c

4h I h

h I 4h I 4h I h I 2h I h I 2h I 2h I

h h h 2h h 2h h h

c b

c c

c c c c c c c

t b t

c c c c c c c

u

u u u

æ _ö÷

ç _÷

ç - _÷ +

ç ÷

è ø

æ ö_÷ æ ö_÷ æ ö_÷

ç _÷ ç _÷ ç _÷

ç + + _÷ +ç - _÷ + -ç - - _÷

ç _÷ ç _÷ ç _÷

ç ÷ ç ÷ ç ÷

è ø è ø è ø



  

(17)

+ - + - - -

-- - + + =  +  +  +  +

1

0

t t t t t *c t *c t t

, x ,x z z ₂ x ,x ₃ x ,x ₂ , ₃ ,

t t c

*c *c t t

1 t 2 t 3

t t t t

z z z z 1 2 3 4

2 3 2 3

:

h 2h h 2h

1 1

M M Q S P H P H

R R h h R h R h

h 2h 2h 6h

R H S R I I I I

R h R h h h

c c

c c c c c

c c t t t t

c c c c c c

t

v v v v

v

qq q q q q q q qq q qq q

q q q q



(11)

æ_- æ öö_÷÷

ç ç _÷÷

ç - + + - ç + - - _÷÷

ç ç _÷÷

ç ç ÷÷

è è øø

æ_- ö_÷ æ_- ö_÷

ç _÷ ç _÷

ç - + + _÷ +ç + _÷

ç _÷ ç _÷

ç ÷ ç ÷

è ø è ø

+ 

 

t 2 t 3 t 4 t 5 t 4 5 6

3

2 3 4 5 2 2 3

c c

t 3 t 4 t 5 t 6 t 4 t 6

2 3 4 5 4 6

c c

0

1 0

h I 2h I 4h I 8h I h 2I 4I 8I

I h

h h h h R h h h

h I 2h I 4h I 8h I 2h I 8h I

h h h h h h

c c c c c c c

c c

c

c c c c c c c

c c c c c

c

c c c c

v

v v æçç_ç- - - ö÷÷_÷_÷

ç ÷

è ø

æ_- ö_÷ æ ö_÷ æ_- ö_÷

ç _÷ ç _÷ ç _÷

ç ç ç

+_ç_ç + _÷÷_÷ +_ç_ç + + _÷÷_÷ +_ç_ç + _÷÷_÷ +

è ø è ø è ø

-

  

t 4 t 5 t 6

4 5 6

2 2 2 2 2

b t 4 b t 6 t 4 t 5 t 6 t b 4 t b 6

4 6 4 5 6 4 6

3 t 4 0 1 c 2 4 1

2h I 8h I 8h I

h h h

h h I 4h h I h I 4h I 4h I h h I 2h h I

h h h h h 2h h

h I 2

2h

c c c

b t

c c c

c c c c c c c

b t b

c c c c c c c

c

v

v v v

æ ö_÷ æ_- ö_÷ æ ö_÷

ç _÷ ç _÷ ç _÷

ç - _÷ +ç + _÷ +ç + + _÷

ç _÷ ç _÷ ç _÷

ç ÷ ç ÷ ç ÷

è ø è ø è ø

3 3 3 3 4 4 4

t 5 t 6 t b 4 t b 6 t 4 t 5 t 6

5 6 2 4 6 3 4 5 6 3

h I 2h I h h I h h I h I h I h I

h h 4h h 4h h h

c c c c c c c

t b t

c c c c c c c

v v v

2

2 2 2 2

, , , ₂ , , ₂ ,

2 2 2 2

0 1 2

2 3 4

2 2

2 2 2

1 2

2

1 1

2

4 2 4 2

4 2 4

4 2

:

t t t t t *c t *c t *c t *c

zz xz x z xz x xz x z z

t t c c c

t t t

t

c c c

c c c c t t t

c c c c c

t t t

c zz zz c c t c c t t w

h h h h

P M R R S R S R

R R h h R h R h

h h h h

M P N M I I I

R h R h h h

h I h

w w I h w I h h

qq q q q q q q

qq qq d - - + + + + + + - - - - + + -+ = + t   

2 2 2 2 2 2

3 4 2 4 2 3 4

0

3 4 2 4 2 3 4

2 2 3 3 3

2 4 2 3

0

4

2 4 2 3

0

1

2

4 4

8 2 8 2 2

c c c c c c

c b t

c c

c

t t t t t t

c

b t b t t t t

c c c c

b

c c c c

h I h I h I h I h I h I

h h h h h h h

h h I h h I h

w w

I h I h I

h h w h h

w

h

æ ö_÷ æ_- ö_÷ æ ö_÷

ç _÷ ç _÷ ç _÷

ç - _÷ +ç + _÷ +ç + + _÷

ç ÷ ç ÷ ç ÷

è ø è ø è ø

æ_- ö_÷

-ç _÷

ç + _÷ + -

-ç _÷ ç + ÷ è ø   

 2 2 2 2 2 4

4 2 4

4 4 4

2 3 4

2 3

1 2

2 4

16 4

16 4 4

c b t b t

c c

t t t

c

t b

c c

t

c c c

h h I h h I

h h

h I h I h

w

I h

w

h h w

æ ö_÷ æ_- ö_÷

ç _÷ ç _÷

ç _÷ +ç + _÷

ç _÷ ç _÷

ç ÷ ç ÷

è ø è ø

æ _ö÷

ç _÷

ç + + ÷

ç _÷ ç ÷ è ø +    (19)

at the bottom face sheet:

1

b b b b b b b

xx,x x , xz ₂ xx,x ₃ xx,x ₂ x, ₃ x,

b

b b b b

b b b 2 b 3 b 4 b 5

xz xz 1 2 3 4

2 3 2 3 4 5

c

0 1 2 3

c c

0

b

:

h 2h h 2h

1

M M Q P H P H

R h h R h R h

2h 6h h I 2h I 4h I 8h I

S R I I I I

h h h h h h

h I

b

c c c c

c c c c c c

c c c c

c c b b b b c

c c c

u u u u

u

q q q q q q

d

+ - + - +

-æ _ö÷

ç _÷

ç

- + = + + + +_ç - - + _÷_÷

ç ÷

è ø

+

    

c c c c

3 b 4 b 5 b 6 b 4 b 5 b 6

2 3 4 5 4 4 6

c c

c c 2 2 2 c

b 4 b 6 b 4 b 5 b 6 t b 4

4

1

6 4 5 6

c

0

0 1

2h I 4h I 8h I 2h I 8h I 8h I

h h h h h h h

2h I 8h I h I 4h I 4h I h h I

h h h h h h

c c c

c b

c c c c c

c c c

t b

c c c c c

u u

æ ö_÷ æ ö_÷

ç _÷ ç _÷

ç - - + _÷ ç - + _÷

ç ÷ ç ÷

è ø è ø

æ ö_÷ æ ö_÷

ç _÷ ç _÷

ç - _÷ +ç - + _÷

-ç _÷ ç _÷

ç ÷ ç ÷

è ø è

+ +

+ ø

 

  t b 6

4 6

3

1

2 2

3 3 2 2 4 4 4

b 4 b 5 b 6 b t 4 b t 6 b 4 b 5 b 6

4 5 6 4 6 4 5 6

c

3 3

b t 4 b 6

4

3

t

4h h I

h

h I 2h I 2h I h h I 2h h I h I h I h I

2h h h 2h h 4h h h

h h I h h I

4h h

c t c

c c c c c c c c

b t b

c c c c c c c

c c

c

u

u u u

æ _ö÷

ç _÷

ç + ÷ +

ç _÷

ç ÷

è ø

æ ö_÷ æ ö_÷ æ ö_÷

ç _÷ ç _÷ ç _÷

ç ç ç

+_ç - + _÷_÷ _ç - _÷_÷ _ç - + _÷_÷

ç ÷ ç ÷ ç ÷

è ø è ø +è ø

-+ + +     3 6 t c u æ _ö÷ ç _÷ ç _÷ ç _÷ ç ÷

è ø

(12)

2

2 2 2 2

b b b b b *c b *c b b

, x ,x z z ₂ x ,x ₃ x ,x ₂ , ₃ ,

b b c

2 2 2 2

*c *c c b b b b

b b b b

z z z z 2 3 4 5

2 3 2 3

c

2 2

b 2

1

b

0 2 3

3 2

:

h h h h

1 1

P P 2S R P H P H

R R 2h h 2R h R h

h h h 3h

R H S R I I I I

2R h R h h h

h I h I

2h

c c

c c c c c

c b b b b

c b

c c c c c

c

v v v v

v

qq q q q q q q qq q qq q

q q q q

+ - + + - + -- - + = + + + + -+    



2 2 2 c c

b 4 b 5 b 4 5 6

3

3 4 5 2 2 3

c

2 2 2 2 c 2 2

b 3 b 4 b 6 b 4 b

0

5 b 6

5

3 3 4 1 4 5 6

2h I 4h I h 2I 4I 8I

I h

h h h 2R h h h

h I h I 2h 2I h I 4h I 4h I

I h

2h h R h h h h

c c c c

c c

c

c c c c c c

c c c c c

c c

c

c c c c c c c

v

æ æ öö_÷÷

ç ç _÷÷

ç - + + ç - - + _÷÷

ç ç _÷÷

ç ç ÷÷

è è øø

æ æ öö_÷÷ æ

ç ç _÷÷ ç

ç - - ç - _÷÷ + - +

ç ç _÷÷

ç ç ÷÷

è è øø è

+ 

 2b 4 2 cb 6

4 6

3 3 3 c 2 2 4 4 4

b 4 b 5 b 6 t b 4 t b 6

0 0

1 1

b 4 b 5 b 6

4 5 6 4 6 4 5 6 2

h I 4h I

h h

h I 2h I 2h I h h I 2h h I h I h I h I

2h h h 2h h 4h h h

c

b t

c c

c c c c c c c

b t b

c c c c c c c c

v v

v v v

ö æ ö

÷ ç ÷

÷ ÷

ç _÷ +ç - _÷

ç _÷ ç _÷

ç ÷_ø ç_è ÷_ø

æ ö_÷ æ_- ö_÷ æ ö_÷

ç _÷ ç _÷ ç _÷

ç - + ÷ +ç + ÷ +ç - + ÷

ç _÷ ç _÷ ç _÷

ç ÷ ç ÷ ç ÷

è ø è ø è ø

+

 

  

2 2 2 2 5 5 c 5 2 3 c 2 3 c

b t 4 b t 6 b 4 b 5 b 6 b t 4 b t 6

4 6 2 4 5 6 3 4 6 3

h h I h h I h I h I h I h h I h h I

4h h 8h 2h 2h 8h 2h

c c c c

t b t

c c c c c c c

v v v

æ ö_÷ æ ö_÷ æ_- ö_÷

ç _÷ ç _÷ ç _÷

ç - _÷ +ç - + _÷ +ç + _÷

ç ÷ ç ÷ ç ÷

è ø è ø è ø

(21)

2

2 2 2 2

b b b b b *c b *c b *c b *c

zz xz,x z, xz,x ₂ xz,x z, ₂ z,

b b c c c c c c

2 2 2 2 2 c 2 c

c c c c b b b

b b b b b 2 b 4

zz zz 2 3 4

2 2 2 4

c c c c c c c c

0 1 2

:

h h h h

1 1

P 2M R R S R S R

R R 4h 2h 4R h 2R h

h h h h h I h I

M P N M I I I

4R h 2R h 4h h 4h h

b b b

w w

qq q q q q q q

qq qq d - - + + - + - + æ-ç + - + - = + + + è + + b   

2 c 2 c 2 c 2 c 2 c 2 c 2 c 3 c 3 c 3 c

b 1 b 2 b 3 b 4 b 2 b 3 b 4 b 2 b 3 b 4

2 3 4 2 3 4 2 3 4

c c c c c c c c c c

0

0 0 1

h I h I h I 2h I h I h I h I h I h I h I

4h 2h h h 4h h h 8h 2h 2h

t

c b b

w

w w w

ö÷÷

ç _÷

ç _÷ø

æ_- ö_÷ æ ö_÷ æ ö_÷

ç _÷ ç _÷ ç _÷

ç + + - ÷ +ç - + ÷ +ç - + ÷

ç _÷ ç _÷ ç _÷

ç ÷ ç ÷ ç ÷

è ø + + ø ø + è è    

2 c 2 c 4 c 4 c 4 c 2 2 c 2 2 c

t b 2 t b 4 b 2 b 3 b 4 b t 2 b t 4

2 4 2 3 4 2 4

c c c c c c c

1 2 2

h h I h h I h I h I h I h h I h h I

8h 2h 16h 4h 4h 16h 4h

t b t

w w w

æ ö_÷ æ ö_÷ æ_- ö_÷

ç _÷ ç _÷ ç _÷

ç ç ç

+_ç - _÷_÷ _ç - + _÷_÷ +_ç + _÷_÷

ç ÷ ç ÷ ç ÷

è ø +è ø è ø

(22)

And at the core:

0

c c

c c c c c c 2 4

xx,x ₂ xx,x x, ₂ x, ₂ xz 0 ₂ ₄

c

c c c c c c

c c c c c c c c

c 3 5 2 3 4 5 2 3

1 ₂ ₄ ₂ ₃ ₄ ₅ ₂ ₃

c c c c c c c

0

1 0

c

8I 16I

4 1 4 8

N P N P S I

R

h R h h h h

12I 32I 2I 4I 8I 16I 2I 4I

I

h h h h h h h h

: c c b u u u u

q q q q

d

æ _ö÷

ç _÷

ç

- + - + =_ç - + _÷_÷

ç ÷

è ø

æ ö_÷ æ ö_÷

ç _÷ ç _÷

ç - + +ç - - + +

+

÷ ÷

ç _÷ ç _÷

ç ÷ ç ÷

è ø è ø

c



  c4 c5

4 5

c c

c c c c c c c c

b 2 b 3 b 4 b 5 t 2 t 3 t 4 t 5

2 3 4 5 2 3 4 5

c c c c c c c c

2 c 2 c 2 c 2 c

b 2 b 3 b 4 b

0

1 1

5

2 3 4 5

c c c c

8I 16I

h h

h I 2h I 4h I 8h I h I 2h I 4h I 8h I

h h h h h h h h

h I h I 2h I 4h I

2h h h h

t b t u u u æ _ö÷ ç _÷

ç - - ÷

ç _÷

ç ÷

è ø

æ ö_÷ æ_- ö_÷

ç _÷ ç _÷

ç - + + _÷ +ç - + + _÷

ç ÷ ç ÷

è ø è ø

+

æ ö

çç - - +

ççè



 

2 c 2 c 2 c 2 c

t 2 t 3 t 4 t 5

2 3 4 5

c c c c

3 c 3 c 3 c 3 c 3 c 3 c 3 c 3 c

b 2 b 3 b 4 b 5 t 2 t 3 t 4 t 5

2 3 4 5 2 3 4 5

c c c c c

2

c c c

2

3 3

h I h I 2h I 4h I

2h h h h

h I h I h I 2h I h I h I h I 2h I

4h 2h h h 4h 2h h h

b t

u u

æ ö

÷ ç ÷

÷ +ç + - - ÷

÷ _ç ÷

÷ ÷

ø è ø

æ ö_÷ æ_- ö_÷

ç _÷ ç _÷

ç - - + _÷ +ç - + + _÷

ç _÷ ç _÷

ç ÷ ç

+

÷

è ø è ø

 

(13)

1

*c *c c c *c *c c c

x ,x ₂ x ,x , ₂ , z ₂ z z ₂ z

c c

c c c c c c

c c c c c c c

c 2 3 5 c 5 6 C 4 6

1 ₂ ₄ ₂ 4 ₂ ₂ 0 2 ₂ ₄

c c c c c c c c c

4 1 4 1 4 12

M H M H S H Q R

R R

h R h R h h

I 8I 16I 8 2I 2I 8I 16I

I I I

R h h R h h h h h

:

c

v v

q q qq q qq q q q q q

d

- + - + - - + =

é æ_ç öù_÷ æ_ç ö_÷

ê + - + - ç_ç - - ÷_÷ú ç_ç - + ÷_÷

ê _ç_è ÷_÷_øú _ç_è ÷_÷_ø

ê ú +

ë û

c

 

c c c c c c c c

3 4 5 6 3 4 5 6

2 3 4 5 2 3 4 5

c c c c c c c c

b 3 b 4 b 5 b 6 t 3 t 4 t 5 t 6

2 3 4 5 2 3 4 5

c c c c c c c c

1

0 0

1

2I 4I 8I 16I 2I 4I 8I 16I

h h h h h h h h

h I 2h I 4h I 8h I h I 2h I 4h I 8h I

h h h h h h h h

c

b t

b

v

v v

v

+

é ù é ù

ê _- _- ₊ ú ₊ê ₊ _- _- ú

ê ú ê ú

ë û ë û

æ ö_÷

æ-ç _÷ ç

ç - - + _÷ +ç - + +

ç _÷

ç ÷

+

è ø è

 



2 c 2 c 2 c 2 c 2 c 2 c 2 c 2 c

b 3 b 4 b 5 b 6 t 3 t 4 t 5 t 6

2 3 4 5 2 3 4 5

c c c c c c c c

3 c 3 c 3 c 3 c

b 3 b 4 b 5 b 6

2 3 4 5

c c c c

1

2 2

3

h I h I 2h I 4h I h I h I 2h I 4h I

2h h h h 2h h h h

h I h I h I 2h I h

4h 2h h h

t

b t

b

v

v v

v

ö÷÷ ÷

ç _÷

ç _÷ø

æ ö_÷ æ ö_÷

ç _÷ ç _÷

ç - + + _÷ +ç + - - _÷

ç ÷ ç ÷

è ø è ø

æ ö_÷

-ç _÷

ç - - + ÷ +

ç _÷

ç _÷ø

+

è



 

 3 ct 3 3 ct 4 3 ct 5 3 ct 6

2 3 4 5

c c c c

3

I h I h I 2h I

4h 2h h h

t

v

æ _ö÷

ç _÷

ç - + + ÷

ç _÷

ç ÷

è ø

(24)

0

*c *c *c *c c c c

xz,x ₂ xz,x z, ₂ z, ₂ ₂ zz

c c

c c c c c c

c c c c c c c c c c

c 2 4 1 2 3 4 1 2 3 4

0 ₂ ₄ ₂ ₃ ₄ ₂ ₃ ₄

c c

c c c c c c

0

c 0

c

:

4 1 4 1 4 8

Q R Q R N P M

R R

h R h R h h

8I 16I I 2I 4I 8I I 2I 4I 8I

I

h h

h h h h h h h h

c c

b

w w

w

q q q q qq qq

- + - - + + =

æ ö_÷ æ_- ö_÷ æ

ç _÷ ç _÷ ç

ç - + _÷ ç + + - _÷ +ç + -

-ç _÷ ç _÷ ç

ç ÷ ç ÷

è ø +è ø è



0t

w

ö÷÷ ÷÷ ç ÷_ø +

c c c c c c c c

b 1 b 2 b 3 b 4 t 1 t 2 t 3 t 4

2 3 4 2 3 4

c c c c c c c c

2 c 2 c 2 c 2 c 2 c 2 c 2

b 1 b 2 b 3 b 4 t 1 t 2 t

2 3 4 2

c c c c c c

1 1

2

h I h I 2h I 4h I h I h I 2h I 4h I

2h h h h 2h h h h

h I h I h I 2h I h I h I h I

4h 2h h h 4h 2h

b t

b

w w

w

æ_- ö_÷ æ_- ö_÷

ç _÷ ç _÷

ç + + - ÷ +ç - + + ÷

ç _÷ ç _÷

ç ÷ ç ÷

è ø è ø

æ_- ö_÷

ç _÷

ç + + - _÷ + +

-ç _÷

ç ÷

è ø

+

 

 c3 2 ct 4

3 4

c

2 c

2h I

h h

t

w

æ _ö÷

ç _÷

ç - _÷

ç _÷

ç ÷

è ø

(25)

where the inertias are given by

(

m

)

n c

i i c c c

c i

z

I z I z

R

i t b m n

2 2

, 1

, 0,1, , 6 , 0,1, , 6

r r

-

-æ _ö÷ ç _÷ = = ç_ç + _÷_÷ çè ø

= = ¼ = ¼

ò

i i

h h

i c

m n

h h

dz dz

,

(26)