Lat. Am. j. solids struct. vol.11 número12

Abstract

This paper dealt with free vibration analysis of thick double curved composite sandwich panels with simply supported or fully clamped boundary conditions based on a new improved higher order sandwich panel theory. The formulation used the first order shear deformation theory for composite face sheets and polynomial description for the displacement field in the core layer which was based on the displacement field of Frostig's second model. The fully dynamic effects of the core layer and face sheets were also considered in this study. Using the Hamilton's principle, the gov-erning equations were derived. Moreover, effects of some im-portant parameters like that of boundary conditions, thickness ratio of the core to panel, radii curvatures and composite lay-up sequences were investigated on free vibration response of the pan-el. The results were validated by those published in the literature and with the FE results obtained by ABAQUS software. It was shown that thicker panels with a thicker core provided greater resistance to resonant vibrations. Also, effect of increasing the core thickness in general was significant decreased fundamental natural frequency values.

Keywords

Free vibration, Double curved sandwich panel, Boundary conditions, Improved higher order sandwich panel theory.

Improved high order free vibration analysis of thick double curved

sandwich panels with transversely flexible cores

1 INTRODUCTION

Structural efficiency is an important attribute for aircraft structures. A higher order theory ap-proach, used by Kant and Patil (1991), replaced sandwich structure with an equivalent higher order shear deformable structure, which lacked the ability to determine local buckling modes and imper-fection effects on the overall behavior. Using the three-dimensional elasticity equations, Bhimaraddi (1993) studied the static response of orthotropic doubly curved shallow shells. He assumed that the ratio of the shell thickness to its middle surface radius is negligible as compared to unity. The

high-K. Malekzadeh Fard a *

M. Livani a

Faramarz Ashenai Ghasemi b

a _{Department of Structural Analysis}

and Simulation, Space Research Insti-tute, Malek Ashtar University of Tech-nology, Tehran-Karaj Highway, Post box: 13445-768, Tehran, Iran m_livani@mut.ac.ir

b _{Department of Mechanical}

Engineer-ing, Shahid Rajaee Teacher Training University (SRTTU), Lavizan, Postal Code 16788-15811,Tehran, Iran. F.a.ghasemi@srttu.edu

*Author email: kmalekzadeh@mut.ac.ir

Latin American Journal of Solids and Structures 11 (2014) 2284-2307

er order sandwich panel theory was developed by Frostig et al. _{(1994, 2004), who considered two}

types of computational models in order to express governing equations of the core layer. The second model assumed a polynomial description of the displacement fields in the core that was based on displacement fields of the first model. Their theory did not impose any restrictions on distribution of the deformation through thickness of the core. Singh (1999) studied free vibration of the open deep sandwich shells made of thin layers and a moderately thick core. Rayleigh–Ritz method was also used to obtain natural frequencies. The improved higher order sandwich plate theory (IHSAPT), applying the first-order shear deformation theory for the face sheets, was introduced by Malekzadeh et al. _{(2005, 2006).}

NOTATIONS

, ,

t c b

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

respec-tively

( , , )

i n

I i =t b c The moments of inertia of the top and bottom face sheets and the core c

z

M Normal bending moments per unit length of the edge of the core

, , ,

i i i i

xy yx xx yy

M M M M Bending and shear moments per unit length of the edge (i=t,b)

* *

, , ,

, , ,

c c c c

nxx nxy nyy nyx

c c c c

nxz nyz nxz nyz

M M M M M M M M

Shear and bending moments per unit length of the edge of the core, (n=1,2,3)

, ,

i i i

xxj yyj xyj

N N N The in-plane external loads in the longitudinal and transverse direction,

respec-tively (i = t, b), (j=1,2)

, , ,

i i i i

xy yx xx yy

N N N N In-plane and shear forces per unit length of the edge (i=t,b)

* *

, , ,

c c c c

xz yz xz yz

N N N N Shear forces per unit length of the edge of the core

, ,

i i i

xxj yyj xyj

N N N In plane resultant forces due to pre stresses (j=t,b)

ij

Q The reduced stiffnesses referred to the principal material coordinates

ij

Q Transformed reduced stiffnesses

,

ix iy

R R Curvature radii of the top face sheet, bottom face sheet and core in x-z and y-z

planes (i=t,b,c) , ,

k k k

u v w Unknowns of the in-plane displacements of the core (k=0,1,2,3)

, ,

c c c

u v w Displacement components of the core

u0i,v0i,w0i Displacement components of the face sheets, (i = t, b)



u_c,

v

w



u_0j,v_0j,w_0j Acceleration components of the face sheets, (j= t, b)

zt , zb ,zc Normal coordinates in the mid-plane of the top and the bottom face sheets and

Latin American Journal of Solids and Structures 11 (2014) 2284-2307

GREEK LETTERS

t,b,c Material densities of the face sheets and the core

_iij _{Normal stress in the face sheets, (i=x,y), j=(t,b)}

iic Normal stress in the core, (i=x,y,z)

xyj,xzj,yzj Shear stress in the face sheets, j=(t,b)

xyc,xzc,yzc Shear stresses in the core

0ixx,0ixy,i0yy,0ixz,0ixz The mid-plane strain components, (i=t,b)

zzc,xxc,yyc Normal strains components of the core layer

xzc,yzc,xyc Shear strains components of the core layer

xi,yi Rotation of the normal section of midsurface of the top face sheet and bottom face

sheet along x and y, respectively(i=t,b)

Zenkour (2005a, b) presented a comprehensive bending, buckling and free vibration analysis of simply supported functionally graded (FG) ceramic–metal sandwich plates. The sandwich plate face sheets were assumed to be isotropic material. Two-constituent material distribution through thick-ness was assumed to vary according to a power law distribution.Garg et al. _{(2006) investigated free}

vibration analysis of simply supported composite and sandwich doubly curved shells. Their formula-tion included Sander's theory based on equivalent single layer approach. Free vibraformula-tion of FG ma-terial sandwich rectangular plates with simply supported and clamped edges was studied by Li et al. _{(2008). The governing equations based on the three-dimensional linear theory of elasticity were}

derived and also they considered two common types of FG sandwich plates, namely sandwich plate with FG face sheets and homogeneous core and sandwich plate with homogeneous face sheets and FG core.

Experimental and analytical investigations of bending and free vibration response of layered FG beams were carried out by Kapuria et al. _{(2008), who demonstrated capability of the zigzag theory}

in modeling mechanics of such beams. Rahmani et al. _{(2009) studied free vibration analysis of open}

single curved composite sandwich panel with a flexible core using a higher order sandwich panel theory. Their formulation used classical shell theory for the face sheets and an elasticity theory for the core layer. Cetkovic and Vuksanovic (2009) investigated global and local responses of laminated and sandwich structures using Reddy's theory, finite element solution and based on equivalent sin-gle layer approach. Biglari and Jafari (2010a) presented a simple three layer theory in order to study vibration and static analysis of open single curved sandwich structures. In their model, they used Donell's theory for the face sheets and considered inconsistent linear stress variation in the core layer. They (2010b) also studied the free vibrations of doubly curved sandwich shell with flexi-ble core based on a refined general-purpose sandwich panel theory. In their theory, the in-plane stresses of the core were neglected.

Free vibration analysis of thick orthotropic plates was performed by Ghugal et al. _{(2011) using a}

Latin American Journal of Solids and Structures 11 (2014) 2284-2307

and bottom surfaces of the plates were satisfied. Rahmani et al. _{(2012) studied the free vibration}

and buckling analyses of circular cylindrical composite sandwich shells subjected to external loads based on the Love–Kirchhoff assumptions for the face sheets. In their theory, the in-plane stresses of the core and the out of stresses of the face sheets were neglected. Mochida et al. _{(2012) studied free}

vibration response of doubly curved shallow shells using approximate Galerkin method. Classical theory of elasticity and von-Karman’s non-linear deformation theory were used by Rafieipour et al. (2013) to investigate free vibration analyses of laminated composite plates.

Using developed four-node quadrilateral element and the zigzag theory, Yasin and Kapuria (2013) studied the static and free vibration analysis of singly- and doubly-curved composite and sandwich shallow shells. In their theory, the transverse normal stresses were neglected. Ghavanloo and Fazelzadeh (2013) examined free vibration analysis of simply supported doubly curved shallow shells. Their formulation was based on Novozhilov's linear shallow shell theory. Using Donell’s non-linear shallow shell theory and Kirchhoff’s hypothesis, Awrejcewicz et al. _{(2013) studied free}

vibra-tion analysis of doubly curved orthotropic shallow shells. Viola et al. _{(2013) used a 2D higher order}

shear deformation theory with nine parameters in order to analyze free vibration analysis of the thick laminated doubly curved shells and panels. Their main assumptions were based on small de-flections and negligible normal stress and strain. A high-order model for the analysis of circular cylindrical composite sandwich shells subjected to low-velocity impact loads was presented by Kha-lili et al. _{(2014). In their theory, the impact behavior of the cylindrical composite sandwich shells}

was described by a high-order sandwich shell theory. Malekzadeh et al. _{(2014) applied the}

first-order shear deformation theory to study effects of some geometrical, physical and material parame-ters on response of the composite plates embedded with shape memory alloy (SMA) wires.

This study investigated free vibration analysis of double curved thick composite sandwich panels using a new improved higher order double curved sandwich panel theory and the second computa-tional model of Frostig (2004). In this work, analytical solution of the displacement field of the core was presented in terms of the polynomials with unknown coefficients according to the second com-putational model of Frostig (2004). Furthermore, the formulation included accurate stress-resultant equations for composite sandwich structures, in which the terms (1+z_c /R_xc)and (1+zc /Ryc)

were imported in equations and exactly integrated. These coefficients could be very important in the structural analysis of thick double curved composite sandwich structures. Simply supported and fully clamped boundary conditions were considered. In order to assure accuracy of the present for-mulations, convergence of the results was examined in details.

2 THEORETICAL FORMULATION

2.1 Basic Assumptions

Latin American Journal of Solids and Structures 11 (2014) 2284-2307

also shown in Figure 1 where t and b refer to the top and bottom face sheets of the panel, respec-tively. Curvature radii of the top face sheet, bottom face sheet and core in x-z and y-z planes are

, ,

tx bx cx

R R R andR_ty, , ,R_by R_cy respectively. The assumptions used in the present analysis were

based on small deformations of linearly elastic materials.

Figure 1: A double curved sandwich panel with laminated face sheets and orthogonal curvilinear coordinates.

2.2 Mathematical Formulation

Mathematical formulation consists of deriving the governing field equations of motion along appro-priate boundary conditions of the face sheets and core. They are derived using the Hamilton's prin-ciple (Frostig 2004).

Using the first shear deformation theory, displacements u, v and w of the face sheets along the x, y (longitudinal) and z (thickness) axes are expressed through the following relations (Reddy 2003):

0

0

0

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

( , , , ) ( , , ) ( , , ) ; ( = , ) ( , , , ) ( , , )

i i

i i

i

i i x

i i y

i

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

v x z y t v x y t z x y t i t b

w x z y t w x y t

y y

= +

= +

=

(1)

wherez_iis vertical coordinate of the each face sheet (i = t, b), measured upward from the mid-plane of each face sheet. Kinematic equations of the face sheets are as follows:

(

)

0 0 0

0 0

, 0, 2 ,

2 , 2 ; ,

i i i i i i i i i i i

xx xx i xx yy yy i yy zz xy xy xy i xy

i i i i i i

xz xz xz yz yz yz

z z z

i t b

e e k e e k e g e e k

g e e g e e

= + = + = = = +

Latin American Journal of Solids and Structures 11 (2014) 2284-2307

where:

0 0 0 0

0 0

0 0 0 0 0 0

0 0 0

, ,

, ,

, ,

i i i i

i i

xx yy

xi yi

i i i i i i

i i i i i

xy xz x yz y

xi yi

i i

i i

y y

i x i i x

xx yy xy

u w v w

x R y R

v u w u w v

x y x R y R

x y x y

 

    

 

 

  

 

   

 

   

       

   

 

 

   

   

(3)

For the thick core layer, displacement fields are based on the second Frostig's model (2004) as fol-lows:

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

2 3

0 1 2 3

2 3

0 1 2 3

2

0 1 2

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

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

, , , , , , , , , .

c c c c

c c c c

xc

c c c c

c c c c

yc

c c c

c c c

z

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

R z

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

R

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

= + + + +

= + + + +

= + +

(4)

Based on small deformations, kinematic relations of the core layer are as follows:

1

(1 )

1

(1 )

1 1

2

(1 ) (1 )

1 2

(1 )

1 2

(1 )

c

c c xx

xc xc

c

c c yy

yc yc

c c

c c

xy xy

xc yc

c c

c c c

xz xz

xc xc

c c

c c c

yz yz

yc yc

u w z R x R

v w z R y R

v u

z R x z R y

w u u

z R x R z

w v v

z R y R z 



 

 

 

 

 _ _  

 

 

 _  _

 _ _

 

  

   

  

  _  _

 _  _ 

  

  _  _

 _  _ 

(5)

Assuming perfect bonding between the top and bottom face sheet–core interfaces, compatibility conditions are as shown below:





 





 





0 0 0

1 1

2 _{1 ;}

2 1

1 2

0 ;

2

k

i i

c ci i x

c ct k

i i

c ci i y

c

i cb

c ci

u z z u h

h For i t k z

v z z v h

h For i b k z

w z z w





 _{   }

  _{ } _{ } 

   

 _{   }   

 

 

  _{    }

 

   _{  }



(6)

Latin American Journal of Solids and Structures 11 (2014) 2284-2307

0

0 0 1

0 0 0

2 ₂ 3 ₃

0

0 0 1

0 0 0

2 ₂ 3 ₃

0 0 0 0

1 2

4

4( ) 2( ) 4

2( ) 4

, ,

4

4( ) 2( ) 4

2( ) 4

, ,

( ) 2( )

,

c

t b t b c c

t x b x c

t b t b c

c t x b x c xc

c c

c

t b t b c c

t y b y c

t b t b c

t y b y yc

c c

c c

t b t b

c c

c

h u

u u h h h u

u u h h u R

u u

h h

h v

v v h h h v

v v h h v R

v v

h h

w w w w

w w

h

y y

y y

y y

y y

- - + -

-+ - +

-= =

- - + -

-+ - +

-= =

- +

= = 0

2

4 .

c c

w

h

-(7)

It can be seen in Equation (7) that the number of unknowns in the core layer is reduced to five. These unknowns are_{u u v v0}c, ₁c, ₀c, ₁c_and

0

c

w . Therefore, generally, all unknowns for a double curved

composite sandwich panel are fifteen as follows:

{

}

The governing equations are derived using the Hamilton's principle which requires that:

0 0

[ ] 0

t t

Ldt K U dt

    





wheredK anddUdenote variation of kinetic energy and strain energy, respectively. Also, t is time

duration between timest₁andt₂, andddenotes variation operator.

The first variation of kinetic energy (assuming homogeneous conditions for displacement and velocity with respect to time coordinate) can be written as follows:

(9)

where

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

c c c i i i

xc yc

z z

dA dx dy dA dx dy i t b

R R

= + + = =

and ( ۟◌ ۟◌ ۟◌ ۟◌ ۟◌ ۟◌ ) denotes the second derivative in time. The first variation of internal potential en-ergy of the sandwich panel is as follows:









,

i

c

i i i i i i t i i i

xx xx yy yy xy xy xz xz yz yz i i t b V

c c c c c c c c c c c c

xx xx yy yy zz zz xy xy xz xz yz yz c V

U dV

dV

          

           



 

 _     _

 

     

 

Latin American Journal of Solids and Structures 11 (2014) 2284-2307

where:

(1 c )(1 c ) , ; ( , )

c c c c c c i i i i i i

xc yc

z z

dV dA dz dx dy dz dV dAdz dx dy dz i t b

R R

= = + + = = =

Using the Hamilton's principle (Equations (8)-(10)) and the kinematic relations (Equations (1)-(7)), equations of motion are obtained as follows:

Q_xzt

R_tx  2 h_c2M2xx,x

c _ 4

h_c3M3xx,x c _ 2

h_c2M2yx,y c _ 4

h_c3M3yx,y c _ 2

R_txh_c2M2xz c _ 4

R_txh_c3M3xz c _

4 h_c2M1xz

*c_12

h_c3M2xz

*c _

I₀t 4I4

c

h_c4 

16I₅c h_c5 

16I₆c h_c6





 



u₀t  4I4

c

h_c4 

16I₆c h_c6





 

 u₀b

2 h_c2(I2

c_ I3

c

R_xc) 4 h_c3(I3

c_ I4

c

R_xc) 8I₄c

h_c4 

8I₅c

h_c4

R_xc  16I₅c

h_c5 

16I₆c

R_xch_c5





 

 u₀c_

2I₃c

h_c2





 4I4

c

h_c3 

8I₅c

h_c4 

16I₆c

h_c5



 u₁c_

I₁t_2htI4

c

h_c4 

8h_tI₅c

h_c5 

8h_tI₆c

h_c6





 __xt _ 2hbI4

c

h_c4 

8h_bI₆c

h_c6





 __xb

(11)

Q_xzb R_bx 

2 h_c2M2xx,x

c _ 4

h_c3M3xx,x c _ 2

h_c2M2yx,y c _ 4

h_c3M3yx,y c _ 2

R_bxh_c2M2xz c _ 4

R_bxh_c3M3xz c

 4 h_c2M1xz

*c_12

h_c3M2xz

*c _ 4I4

c

h_c4 

16I₆c

h_c6





 _u₀t_

I₀b_4I4

c

h_c4 

16I₅c

h_c5 

16I₆c

h_c6





 _u₀b_

2 h_c2(I2

c I3

c

R_xc) 4 h_c3(I3

c I4

c

R_xc) 8I₄c

h_c4 

8I₅c

h_c4

R_xc 16I₅c

h_c5 

16I₆c

R_xch_c5





 _u₀c

2I₃c

h_c2





 4I4

c

h_c3 

8I₅c

h_c4 

16I₆c

h_c5





u₁c 2htI4

c

h_c4 

8h_tI₆c

h_c6





 

 _xt _

I₁b2hbI4

c

h_c4 

8h_bI₅c

h_c5 

8h_bI₆c

h_c6





 

 _xb

(12)

Q_yzt R_ty 

2 h_c2M2yy,y

c _ 4

h_c3M3yy,y c _ 2

h_c2M2xy,x c _ 4

h_c3M3xy,x c _ 2

R_cyh_c2M2yz c _ 4

R_cyh_c3M3yz c _ 4

h_c2M1yz

*c

12 h_c3M2yz

*c 

I₀t4I4

c

h_c4 

16I₅c

h_c5 

16I₆c

h_c6





 _v₀t 4I4

c

h_c4 

16I₆c

h_c6





 _v₀b

2 h_c2(I2

c_ I3

c

R_yc) 4 h_c3(I3

c_ I4

c

R_yc) 8I₄c

h_c4 

8I₅c

h_c4

R_yc  16I₅c

h_c5 

16I₆c

R_ych_c5

       v0

c_

2I₃c

h_c2





 4I4

c

h_c3 

8I₅c

h_c4 

16I₆c

h_c5



 v₁c_

I₁t_2htI4

c

h_c4 

8h_tI₅c

h_c5 

8h_tI₆c

h_c6





 _y

t _ 2hbI4

c

h_c4 

8h_bI₆c

h_c6





 _y b

Latin American Journal of Solids and Structures 11 (2014) 2284-2307

Q_yzb R_by 

2 h_c2M2yy,y

c _ 4

h_c3M3yy,y c _ 2

h_c2M2xy,x c _ 4

h_c3M3xy,x c _ 2

R_cyh_c2M2yz c _ 4

R_cyh_c3M3yz c

 4 h_c2M1yz

*c 12

h_c3M2yz

*c  4I4

c

h_c4 

16I₆c h_c6





 

 v₀t

I₀b4I4

c

h_c4 

16I₅c h_c5 

16I₆c h_c6





 

 v₀b

2 h_c2(I2

c_ I3

c

R_yc) 4 h_c3(I3

c_ I4

c

R_yc) 8I₄c

h_c4 

8I₅c

h_c4

R_yc 16I₅c

h_c5 

16I₆c

R_ych_c5

       v0

c_

2I₃c

h_c2





 4I4

c

h_c3 

8I₅c

h_c4 

16I₆c

h_c5





v₁c_{ }2htI4

c

h_c4 

8h_tI₆c

h_c6





 _y t _

I₁b_2hbI4

c

h_c4 

8h_bI₅c

h_c5 

8h_bI₆c

h_c6





 _y b

(14)

Q_xzt_,x

Q_yzt _,y₍Nxx t

R_tx  N_yyt

R_ty ) R_zc

h_c  1 R_cxh_c M1xx

c  4

h_c2Mz c 2

R_cxh_c2M2xx c  1

R_cyh_cM1yy c

 2 R_cyh_c2M2yy

c _ 1

h_cM1xz,x c _ 2

h_c2M2xz,x c _ 1

h_cM1yz,y c _ 2

h_c2M2yz,y c

 I₀t_I2

c

h_c2

4I₃c

h_c3 

4I₄c

h_c4





 



w₀t_{ }I2

c

h_c2

4I₄c

h_c4





 

 w₀b_ I1

c

h_c  2I₂c

h_c2 

4I₃c

h_c3 

8I₄c

h_c4





 

 w₀c

(15)

Q_xzb_,xQ_yzb_,y(Nxx b

R_bx  N_yyb

R_by ) R_zc

h_c  1 R_cxh_cM1xx

c _ 4 h_c2Mz

c_ 2 R_cxh_c2M2xx

c _ 1 R_cyh_cM1yy

c _

2 R_cyh_c2M2yy

c _ 1 h_cM1xz,x

c _ 2 h_c2M2xz,x

c _ 1 h_cM1yz,y

c _ 2 h_c2M2yz,y

c _{ }I2

c

h_c2 

4I₄c h_c4

 

 

 w₀t 

I₀b_I2

c

h_c2 

4I₃c h_c3 

4I₄c h_c4

 

 



w₀b_{ }I1

c

h_c  2I₂c

h_c2 

4I₃c h_c3 

8I₄c h_c4

 

 

 w₀c

(16)

M_xxt_,x M_xyt _,y

Q_xzt _ht h_c2M2xx,x

c _2ht h_c3 M3xx,x

c _ ht h_c2M2yx,y

c _2ht h_c3 M3yx,y

c _ ht R_cxh_c2M2xz

c

 2ht R_cxh_c2M3xz

c _2ht h_c2 M1xz

*c _6ht h_c3 M2xz

*c _

I₁t _2htI4

c

h_c4 

 

 8htI5

c

h_c5 

8h_tI₆c h_c6

  u₀t_

(2htI4

c

h_c4 

8h_tI₆c h_c6 )u0

b

 ht h_c2(I2

c_ I3

c

R_xc) 2h_t

h_c3 (I3

c_ I4

c

R_xc) 4h_tI₄c

h_c4 

4h_tI₅c R_xch_c4

8h_tI₅c h_c5 

8h_tI₆c R_xch_c5

 

 

 u₀c_

htI3

c

h_c2 

2h_tI₄c h_c3 

4h_tI₅c h_c4 

8h_tI₆c h_c5

 

 



u₁c_{ }hthbI4

c

h_c4 

4h_th_bI₆c h_c6

 

 

 _xb_

I₂t_ht

2

I₄c h_c4 

4h_t2

I₅c h_c5 

4h_t2

I₆c h_c6

 

 

 _xt

Latin American Journal of Solids and Structures 11 (2014) 2284-2307

M_xxb_,x M_xyb_,y

Q_xzb _hb h_c2M2xx,x

c _2hb h_c3 M3xx,x

c _hb h_c2M2yx,y

c _2hb h_c3 M3yx,y

c _ hb R_cxh_c2M2xz

c _

2h_b R_cxh_c2M3xz

c _2hb h_c2 M1xz

*c _6hb h_c3 M2xz

*c _ 2htI4

c

h_c4

 

 8htI6

c

h_c6

  u₀t_(

I₁b_2hbI4

c

h_c4 

8h_bI₅c h_c5 

8h_bI₆c h_c6 )u0

b_

h_b h_c2(I2

c_ I3

c

R_xc) 2h_b

h_c3 (I3

c_ I4

c

R_xc) 4h_bI₄c

h_c4 

4h_bI₅c R_xch_c4

8h_bI₅c h_c5 

8h_bI₆c R_xch_c5

 

 _u₀c_

h_bI₃c h_c2 

2h_bI₄c h_c3 

4h_bI₅c h_c4 

8h_bI₆c h_c5

 

 _u₁c_{ }hthbI4

c

h_c4 

4h_th_bI₆c h_c6

 

 __xt_ I₂b_hb

2

I₄c h_c4 

4h_b2

I₅c h_c5 

4h_b2

I₆c h_c6

 

 __xb

(18)

M_xyt_,xM_yyt _,yQ_yzt ht h_c2M2yy,y

c _2ht h_c3 M3yy,y

c _ ht h_c2M2xy,x

c _2ht h_c3 M3xy,x

c _ ht R_cxh_c2M2yz

c _

2h_t R_cxh_c2M3yz

c _2ht h_c2 M1yz

*c _6ht h_c3 M2yz

*c _

I₁t2htI4

c

h_c4 

 

 8htI5

c

h_c5 

8h_tI₆c h_c6

 

v₀t 2htI4

c

h_c4 

8h_tI₆c h_c6

 

 

 v₀b

ht h_c2(I2

c_ I3

c

R_yc) 2h_t

h_c3 (I3

c_ I4

c

R_yc) 4h_tI₄c

h_c4 

4h_tI₅c R_ych_c4

8h_tI₅c h_c5 

8h_tI₆c R_ych_c5

       v0

c_

htI3

c

h_c2 

2h_tI₄c h_c3 

4h_tI₅c h_c4 

8h_tI₆c h_c5

 

 

 v₁c_

I₂t_ht

2

I₄c h_c4 

4h_t2

I₅c h_c5 

4h_t2

I₆c h_c6

 

 



t_{y }hthbI4

c

h_c4 

4h_th_bI₆c h_c6

 

 

 b_y

(19)

M_xyb_,x

M_yyb_,y

Q_yzb _hb

h_c2M2yy,y c _2hb

h_c3 M3yy,y c _hb

h_c2M2xy,x c _2hb

h_c3 M3xy,x c _ hb

R_cyh_c2M2yz c _ 2hb

R_cyh_c2M3yz c

2hb

h_c2 M1yz

*c _6hb

h_c3 M2yz

*c _ 2htI4

c

h_c4





 8htI6

c

h_c6





v₀t  I₁b2hbI4

c

h_c4 

8h_bI₅c

h_c5 

8h_bI₆c

h_c6





 

 v₀b

h_b h_c2(I2

c_ I3

c

R_yc) 2h_b

h_c3 (I3

c_ I4

c

R_yc) 4h_bI₄c

h_c4 

4h_bI₅c

R_ych_c4 

8h_bI₅c

h_c5 

8h_bI₆c

R_ych_c5

       v0

c_

h_bI₃c h_c2 

2h_bI₄c h_c3 

4h_bI₅c h_c4 

8h_bI₆c h_c5





 _v₁c hthbI4

c

h_c4 

4h_th_bI₆c h_c6





 _y t 

I₂bhb

2

I₄c h_c4 

4h_b2

I₅c h_c5 

4h_b2

I₆c h_c6





 _y b

Latin American Journal of Solids and Structures 11 (2014) 2284-2307

(N_xxc_,x

N_yxc_,y 4

h_c2M2xx,x c  4

R_cxh_c2M3xx,x c  1

R_cx M1yx,y c  4

h_c2M2yx,y c  4

R_cxh_c2M3yx,y c  1

R_cx Nxz

c 

4 R_cxh_c2M2xz

c _ 4

R_cx2

h_c2M3xz c _ 8

h_c2M1xz

*c _ 12

R_cxh_c2M2xz

*c _ 1

R_cx Nxz

*c

) I₀c_ I2

c

R_xc2 

2I₁c

R_xc  8 h_c2(I2

c_ I3

c

R_xc) 

 

 8 h_c2

R_xc(I3

c_ I4

c

R_xc) 16I₄c

h_c4 

32I₅c

R_xch_c4

16I₆c

R_xc2

h_c4



 u₀c_

I₁c_ I2

c

R_xc  8I₃c

h_c2 

8I₄c

h_c2

R_xc  16I₅c

h_c4 

16I₆c

R_xch_c4





 _u₁c

 2 h_c2(I2

c I3

c

R_xc) 4 h_c3(I3

c I4

c

R_xc) 8I₄c

h_c4  16I₅c

h_c5  8I₅c R_xch_c4

16I₆c R_xch_c5 



 _u₀b₍ 2

h_c2(I2

c I3

c

R_xc) 4 h_c3(I3

c I4

c

R_xc)

8I4

c

h_c4 

16I₅c h_c5 

8I₅c R_xch_c4

16I₆c R_xch_c5)u0

t_ hb

h_c2(I2

c_ I3

c

R_xc) 2h_b

h_c3 (I3

c_ I4

c

R_xc) 4h_bI₄c

h_c4 

8h_bI₅c h_c5 

4h_ch_bI₅c R_xch_c5 



 

8h_ch_bI₆c

R_xch_c6





_xb_{ }ht

h_c2(I2

c_ I3

c

R_xc) 2h_t

h_c3 (I3

c_ I4

c

R_xc) 4h_tI₄c

h_c4 

8h_tI₅c

h_c5 





 4hchtI5

c

R_xch_c5 

8h_ch_tI₆c

R_xch_c6



 _xt

(21)

M₁c_xx,x

N_xz*c 4

h_c2M3xx,x

c 

M₁c_yx,y 4

h_c2M3yx,y

c  1

R_cxM1xz c  4

R_cxh_c2M3xz

c 12

h_c2M2xz

*c 

2I₃c

h_c2 

4I₄c

h_c3 

8I₅c

h_c4 

16I₆c

h_c5





 



u₀t_ 2I3

c

h_c2 

4I₄c

h_c3 

8I₅c

h_c4 

16I₆c

h_c5





 

 u₀b_

I₁c_ I2

c

R_xc  8I₃c

h_c2 

8I₄c

h_c2

R_xc  16I₅c

h_c4 

16I₆c

R_xch_c4





 _u₀c_

I₂c_8I4

c

h_c2 

16I₆c

h_c4





 _u₁c_

htI3

c

h_c2 

2h_tI₄c

h_c3 

4h_tI₅c

h_c4 

8h_tI₆c

h_c5





 __xt  hbI3

c

h_c2





 2hbI4

c

h_c3 

4h_bI₅c

h_c4 

8h_bI₆c

h_c5



 _xb

Latin American Journal of Solids and Structures 11 (2014) 2284-2307

In Equations (11)-(25), c

(

)

n n

I = ¼ are moments of inertia for the core layer, as indicated

below:

2

2

(1 )(1 ) ; 0,1,..., 6

c

c

h

c n c c

n c c c

h xc yc

z z

I z dz n

R R









N_yyc_,yN_xyc_,x 4 h_c2M2yy,y

c _ 4

R_cyh_c2M3yy,y c _ 1

R_cy M1xy,x

c _ 4

h_c2M2xy,x c _ 4

R_cyh_c2M3xy,x c

 1 R_cy Nyz

c _ 4

R_cyh_c2M2yz c _ 4

R_cy2

h_c2M3yz c _ 8

h_c2M1yz

*c _ 12

R_cyh_c2M2yz

*c _ 1

R_cy Nyz

*c_

2

h_c2(I2

c_ I3

c

R_yc)

4

h_c3(I3

c_ I4

c

R_yc)

8I₄c

h_c4 

16I₅c

h_c5 

8I₅c

R_ych_c4

16I₆c

R_ych_c5

       v0

t _

2

h_c2(I2

c_ I3

c

R_yc)

4

h_c3(I3

c_ I4

c

R_yc)

8I₄c

h_c4 

16I₅c

h_c5 

8I₅c

R_ych_c4

16I₆c

R_ych_c5

       v0

b_

I₀c_ I2

c

R_yc2 

2I₁c

R_yc 

8

h_c2(I2

c_ I3

c

R_yc) 

 

 _h 8

c

2

R_yc(I3

c_ I4

c

R_yc)

16I₄c

h_c4 

32I₅c

R_ych_c4

16I₆c

R_yc2

h_c4



  v0

c_

I₁c_ I2

c

R_yc

8I₃c

h_c2 

8I₄c

h_c2

R_yc

16I₅c

h_c4 

16I₆c

R_ych_c4

       v1

c_

ht

h_c2(I2

c_ I3

c

R_yc)

2h_t

h_c3 (I3

c_ I4

c

R_yc)

4h_tI₄c

h_c4 

8h_tI₅c

h_c5 





 4hchtI5

c

R_ych_c5 

8h_ch_tI₆c

R_ych_c6



  x

t _

h_b h_c2(I2

c_ I3

c

R_yc)

2h_b

h_c3 (I3

c_ I4

c

R_yc)

4h_bI₄c

h_c4 

8h_bI₅c

h_c5 

4h_ch_bI₅c

R_ych_c5 



 

 8hchbI6

c

R_cyh_c6



  y

b

(23)

M₁c_yy,y

N_yz*c_ 4

h_c2M3yy,y c _

M₁c_xy,x 4

h_c2M3xy,x c _ 1

R_cy M1yz c _ 4

R_cyh_c2M3yz c _

12

h_c2M2yz

*c _ 2I3

c

h_c2 

4I₄c

h_c3 

8I₅c

h_c4 

16I₆c

h_c5





 _v₀t_ 2I3

c

h_c2 

4I₄c

h_c3 

8I₅c

h_c4 

16I₆c

h_c5





 _v₀b_

I₁c_ I2

c

R_yc

8I₃c

h_c2 

8I₄c

h_c2

R_yc

16I₅c

h_c4 

16I₆c

R_ych_c4

       v0

c_

I₂c_8I4

c

h_c2 

16I₆c

h_c4





 _v₁c_

htI3

c

h_c2 

2h_tI₄c

h_c3 

4h_tI₅c

h_c4 

8h_tI₆c

h_c5





 _t_y hbI3

c

h_c2





 2hbI4

c

h_c3 

4h_bI₅c

h_c4 

8h_bI₆c

h_c5



 _yb

(24)

N_xzc_,x

N_yzc_,y 8 h_c2Mz

c_ 1 R_xc Nxx

c _ 4 R_xch_c2M2xx

c _ 1 R_yc Nyy

c _ 4 R_ych_c2M2yy

c _ 4 h_c2M2xz,x

c _

4 h_c2M2yz,y

c _ I1

c

h_c  2I₂c

h_c2 

4I₃c h_c3 

8I₄c h_c4

 

 _w₀t_{ }I1

c

h_c  2I₂c

h_c2 

4I₃c h_c3 

8I₄c h_c4

 

 _w₀b_

I₀c_8I2

c

h_c2 

16I₄c h_c4

 

 _w₀c