Lat. Am. j. solids struct. vol.12 número7

Abstract

The available shear correction factors have mainly been proposed for single-layer rectangular plates with zero shear tractions and isotropic homogenous materials. The present analytical shear factors are especially suitable for the first-order zigzag or layerwise theories of circular sandwich plates with functionally graded cores/face sheets and simultaneous normal and shear tractions. Although the present layerwise correction factors are general, they are evaluated for the modal analyses where effects of the shear correction are more remarkable than those of the stress analyses. It is the first time that the concept of the local shear correction factor is introduced. To present more accurate results, the Mori-Tanaka micromechanical-based material properties model is used instead of the traditional rule of mixtures. The governing equa-tions are solved using the analytical Taylor transform method. Comparisons made among results associated with the known shear correction factors, present results, and results of the three-dimensional theory of elasticity reveal that significant enhance-ments may occur through using the proposed analytical shear correction factors.

Keywords

Local shear correction factors; analytical Taylor transform solu-tion; functionally graded circular sandwich plate; zigzag theory; free vibration.

Novel Layerwise Shear Correction Factors

for Zigzag Theories of Circular Sandwich

Plates with Functionally Graded Layers

M. Shariyat a M.M. Alipour b

a _{Professor, Faculty of Mechanical}

Engi-neering, K.N. Toosi University of Tech-nology, Tehran, Iran.

Email: m_shariyat@yahoo.com (Corresponding author)

b _{Ph.D., Faculty of Mechanical}

Engineer-ing, K.N. Toosi University of Technolo-gy, Tehran, Iran.

Email: m.mollaalipur@gmail.com

http://dx.doi.org/10.1590/1679-78251477

Latin A m erican Journal of Solids and Structures 12 (2015) 1362-1396 1 INTRODUCTION

Three-layer sandwich plates with protective or load carrying face sheets and spacer, damper, or stiffener cores have extensively been employed in various engineering applications and hi-tech struc-tures. Various requirements may be covered through using different combinations of the stiff or soft materials for the face sheets and cores. Employing functionally graded face sheets or cores may re-duce or eliminate the discontinuities in the distribution of the transverse stresses at the interfaces between the layers.

Although the three-dimensional theory of elasticity is an accurate approach this theory cannot be directly applied with ease for multilayer plates with arbitrary variations of the material proper-ties, non-uniform tractions, and general geometries and boundary conditions, especially when thick-ness of the layers is small. Majority of the researchers have used equivalent single layer theories (Reddy, 2005; Shariyat, 2009a; Shariyat, 2009b; Ebrahimi et al., 2009; Mantari and Oktem, 2011; Sofiyev, 2014) for analysis of the multilayer plates. It is evident that accuracy of these theories may encounter problems when number of the layers increases or when the material properties experience severe changes in the transverse direction. In such cases, using the layerwise or zigzag theories is more appropriate (Pandit et al., 2008; Fares and Elmarghany, 2008; Shariyat, 2010; Shariyat, 2012a). For the both general categories of the plates, i.e., the equivalent single-layer and zigzag first-order theories, the first-order shear-deformation theories (FSDT) are computationally more economic and relatively simpler. For these reasons, they have been attractive for performing the stress and bending (Alipour, and Shariyat, 2010; Reddy et al., 1999), vibration (Alipour et al., 2010; Alipour and Shariyat, 2011a), and buckling (Shariyat and Alipour, 2013a; Alipour and Shariyat, 2013) analyses of plates under complicated conditions. Nevertheless, based on this type of theories, the transverse shear strains become constant in the thickness direction; so that the shear correction factors have to be used to correct the strain energy of the transverse shear stresses and consequent-ly, adjust the transverse shear stiffness of the plate.

Ste-Latin A m erican Journal of Solids and S tructures 12 (2015) 1362-1396

Latin A m erican Journal of Solids and Structures 12 (2015) 1362-1396 2 DESCRIPTION OF VARIATIONS OF THE MATERIAL PROPERTIES

AND THE ZIGZAG DISPLACEMENT FIELD

Geometric parameters of the circular sandwich plate with functionally graded layers are shown in Fig.1. It is assumed that each layer of the sandwich plate is generally made of a mixture of ce-ramic and metallic constituent materials. Using a power–law for description of variations of the volume fraction of the metallic constituent material in the transverse direction, one may write:

g m

h

z

V











 



2

1

(1) So that:

1

  _m c V

V ₍₂₎

where the subscripts c and m stand for ceramic and metal, respectively and g is the positive defi-nite volume fraction index. Therefore, Vc and Vm are the volume fractions of the ceramic and

metallic constituent materials, respectively. According to Mori-Tanaka material properties model, variations of the bulk and shear moduli can be represented by:









4 3

-

_,

--

₁

1--

_,

(9

8 )

--

_{1 1-}

6(

2 )

c m

m c

m c

m

c c

c m c c c

c

m c

m c c c

m

c c

V

V

G

G G

V

G

G

f

G

G

G

G

_V

G

G

f

 



_



_

_

_

_





_{ }





_{ }



_

_



_

 



_



(3)

from which Young’s modulus and Poisson’s ratio may be determined as:

9 _, 3 - 2

3 6 2

G G

E v

G G

 

 

    (4)

Latin A m erican Journal of Solids and S tructures 12 (2015) 1362-1396

Therefore, variations of the material properties within each layer may be determined from:

Latin A m erican Journal of Solids and Structures 12 (2015) 1362-1396 Although the traditional rule of mixtures is generally not appropriate for determination of the elastic moduli and Poisson’s ratio, its application may be justified for modeling variations of the mass density within each layer (Shariyat and Jafari, 2013):

cVc mVm

 

 



₍₆₎

To enable accurately tracing the layerwise variations of the in-plane displacement component, a zigzag theory that is a result of a superposition of linear layerwise and linear global fields is adopted. So that variations of the resulting displacement field may be expressed as follows within each layer:

( ) ( )

0

( ,z, ) ( , , ) ( , , ) ,

i i

g l

u r t u r z t u r z t ww ₍₇₎

where u and w are the radial and lateral displacement components. The subscripts g, l, and 0 denote the global and local components and value of the displacement component at the reference plate, respectively. The z coordinate is measured from the reference layer (e.g., the mid-plane of the plate) and positive upward. Imposing the interlaminar kinematic continuity conditions, the resulting displacement field of the sandwich plate may be expressed as follows:

(1) (2)

2 2 2 2

0 1

(2) 2 2

0

(3) (2)

2 2 2 2

0 3

2 2

0 3 1

,

2 2 2 2

,

2 2

,

2 2 2 2

,

2 2

g l l

g l

g l l

h h h h

u z z z h

h h

u u z z z

h h h h

u z z h z

h h

w w h z h

















 _{ } _  _{   }

 

 _{ }

 

_     



 _{ } _  _{     }

 _ _



     

(8)

where



_g and ( )i _{,( 1,2,3)} l i



 are the global rotation of the radial cross section and local rotations of the individual layers, respectively. h1, h3, and h2 are thicknesses of the face sheets and core,

respectively (Fig. 1). Eq. 8 may be rewritten as:

(1) (2)

2 2 2 2

0 1

(2) 2 2

0

(3) (2)

2 2 2 2

0 3

2 2

0 3 1

,

2 2 2 2

,

2 2

,

2 2 2 2

,

2 2

h h h h

u z z h

h h

u u z z

h h h h

u z h z

h h

w w h z h











 _ _  _{   }

 

 _{ }

 

_    



 _ _  _{     }

 _ _



     

Latin A m erican Journal of Solids and S tructures 12 (2015) 1362-1396 where

( )i ( )i

g l











₍₁₀₎

3 DERIVATION OF THE LOCAL SHEAR CORRECTION FACTOR (LSCF)

In the present section, the local shear correction factors are derived for functionally graded circu-lar sandwich plates with normal and shear tractions on the top and bottom surfaces. Although power-law variations are adopted for the volume fractions in the preceding section, present formu-lations may similarly be employed for any type of variations of the material properties.

For small deflections of axisymmetric circulate plates, the strain-displacement relations in the polar coordinate may be written as:

Since magnitude of the transverse normal strain is usually ignorable in comparison with that of the in-plane strains, the stress-strain relation of the ith layer may have the following form:

( ) ( ) ( ) ( )

11 12

( ) ( ) ( ) ( )

21 22 ( )

( ) ( )

33

0

0

( 1,2,3)

0

0

i _{i i} i

r r

i i i i

i

i i

rz rz

C

C

C

C

i

C

 

















_

_









_

_













_

_









_

_

















(12)

where:

 

 

   

 

 

 

( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

( ) ( ) ( ) ( ) ( )

11 22 _{( ) ( )} 2 , 12 21 _{( ) ( )} 2 , 33 _{( ) ( )}

2 1

1 1

i i i i i i i

i

i i i i i

i i

i i i i

E z v z E z E z

C C C C C

v z

v z v z

    

  

   

_{ } _{   } (13)

Based on the three-dimensional theory of elasticity, the equilibrium equations of the axisymmetric circular plate may be expressed as:

r

r rz _u

r r z













_

 _{ } _

  (14)

w z r

r r

z

rz









    

( )

1

(15) where



r,



and



z are the normal stresses and



rz is the transverse shear stress. Although the radial and transverse inertia body forces (



u and



w) appeared in Eqs. (14,15) may affect the shear correction factor, their effects may be ignored in comparison to those of the transversely applied distributed tractions, especially for the thin plates. This assumption has been employed in derivation of all of the available shear correction factors.

, ,

r rz

u u u w

r  r z r



 







  

Latin A m erican Journal of Solids and Structures 12 (2015) 1362-1396 Based on Eq. 9, local variations of the in-plane displacement component of the ith layer may be rewritten as:

( ) ( ) ( ) ( ) ( )

0 ; _{2 2}

i i i i i i i

l

h h

u u 

 

 



 ₍₁₆₎

where



( )i

are the local z-coordinates of the layers that are measured from the mid-plane of the corresponding layer and are positive upward.

Based on Eqs. (11,12,16), the in-plane stress components will be:







 



( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

11 0, , 0

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

11 0, , 1 0

i

i i i i i i i i

r r l r l

i i i i i i i i i

r l r l

v

C u u

r

C v u u

r





 

 



 

 

  

   

  

 _{ }



 

 _{   }

 

 _{ }



(17)

By substituting Eq. (17) into Eq. (14), the radial equilibrium equation of each layer of the sand-wich plate may be expressed based on the present zigzag theory as:





( ) 2

( ) 2 ( ) ( ) ( ) 2

11 0 ( ) 2 2

1 1

0,

i

i i i i rz

l i

C u

r r

r r



 



 

  

     _   _



 _ _ (18)

By integration of the elasticity equilibrium Eq. (18) across the plate thickness, the transverse shear stress within each layer may be obtained as:

( )

( ) ( ) 2 ( ) ( ) 2 ( ) ( ) ( ) ( )

0 _{( )} 11 _{( )}

1 ,

i

i i i i i i i i

rz l i i

X

X u Y C C d

Y









   

   

     _{ }  _{ }

 

   

 



(19)

( )i

C is an integration constant. Integrating the transverse shear stress across the thickness of

each layer, the transverse shear force per unit length of each layer can be obtained:

/2

( ) ( ) ( ) /2

i

i

h

i i i

r rz

h

Q



d









₍₂₀₎

Latin A m erican Journal of Solids and S tructures 12 (2015) 1362-1396 1 2 3 2 1 3 1 2 3 2 (1) (2) 2 2 (3) (2) 2 2 (1) 2 (3) 2 (1) (2) 2 2 (3) (2) 2 2 h h h h

rz h st

rz h sb

rz h rz h

rz h rz h

u

u

u

u

q

q

         













 _   _           _   _      _{ }    _       _       _   _           _   _      _{ }  

















_















_







_



(21)

where

q

_stand

q

_sbare the shear tractions of the top and bottom surfaces of the sandwich plate. Therefore:

2

(1) (2) (1) (1)

2

, , ,

st rz h sb

C q C C q





_{ }         (22) Using the Eqs. (20,21), the unknown displacement parameters _u0( )i _{and ,} ( )i

l



may be obtained based on the shear force of the relevant layer per unit length. On the other hand, based on Eq. (15) and the boundary and continuity conditions (at the interfaces between layers):

3 2 2 2 /2 (1) (1) /2 (2) (2) (1) 2 /2 ₍₃₎ (3)

(

)

1

(

)

1

(

)

1

h rz z nt z h rz

z z h

z h rz z nb z

r

d

q

r

r

r

d

r

r

r

d

q

r

r













 







 _      































(23) nt

Latin A m erican Journal of Solids and Structures 12 (2015) 1362-1396



























(1) 3

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

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

3 2 2 2 2 3 2 2

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

2 (1) 2

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

3 1 _{8 2}

4 3 4

6 6 4 2

8 8 3

rz h h h h Q h h Q h h Q

h h h h h h h h h

h h h Q h Q h h h Q h h h h Q h Q h h Q

h h h Q h h h h h

                                      _      _ 











2

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

2 2 2 (1)2

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

4

4 3 6 12 st

Q h Q h h Q h h Q

h h h Q h Q h h Q q

                  _ 



























(2) 2 2 2

2 2 2 1 3 1 1 3 3

2

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

3 2 2 4 2 3 2 2 2

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

2 2 2 2 2 2 3

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

3 1 ₆

4 3 4

2 8 4

12 8 8

rz h Q h h h h

h h h h h h h h h

h h Q h Q h h Q h Q h h h Q h h h h Q

h h Q h Q h Q h h h h h h Q

                                           



















2 (2)

3 3 1

2 2 2 2 2 2 2 (2)2

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

(1)

2

12 8 24 32

rz h

h Q

h h Q h Q Q h h h h h h Q h Q h h h Q

            _{ }             _       _  (24)



























(3) 3

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

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

2 3 2 2 2 3 2 2

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

2 (3) 2

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

3 1 _{8 2}

4 3 4

6 6 4 2

8 8 3

rz h h h h Q h h Q h h Q

h h h h h h h h h

h h h Q h Q h h h Q h h h h Q h h Q h Q

h h h Q h h h h h

                                      _      _ 











2

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

2 2 2 (3)2

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

4

4 3 6 12 sb

Q h Q h h Q h h Q

h h h Q h Q h h Q q

                  _  where ( ) ( )2 1 i i i i E v



  .

On the other hand, based on the present first-order zigzag theory, the constitutive-equations-based transverse shear stress of each layer is:





( )i ( )i ₍ ( )i ₎ ( )i _,

rz G l wr



 





 ₍₂₅₎

Latin A m erican Journal of Solids and S tructures 12 (2015) 1362-1396





/2 /2

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

/2 /2

, ,

( )

i i

i i

h h

i i i i i i i i i i

r rz l r

h h

Q



d





L



w

G



d



 















₍₂₆₎

where



(i)

is the local shear correction factor of the ith layer. Based on Eq. (26), Eq. (25) may be rewritten in terms of the shear force per length as:

( 27 )

 

( ) ( )

( ) ( )

( )

i i

i i

rz i r

G

Q

   

The strain energy associated with the elasticity-based transverse shear stress (19) can be deter-mined from the following equation:

 

/2

( ) ( )2 ( )

( ) ( )

0 /2

1

1

₂

2

i

i

h b

i i i

s _{i i} rz

h

U

r drd

G













 

₍₂₈₎

On the other hand, the strain energy associated with the constitutive-equations-based transverse shear stress maybe corrected as follows to match Eq. (28):

 

( ) ( ) /2

( ) ( )2 ( )

( ) ( )2 0 /2

1

₂

2

i

i

i i

h b

i i i

s i i r

h

G

U



Q



rdrd









 

₍₂₉₎

Equating the strain energy expressions appeared in Eqs. (28,29) leads to the following set of new general local correction factors for the three-layer functionally graded sandwich plate.

 

 

( ) ( ) /2

( )2 ( ) ( )2

0 /2 ( )

/2

( )2 ( ) ( ) ( )

0 /2

2

1 ₂

i

i

i

i

i i

h b

i i

r i h i

h b

i i

rz

i i

h G

Q r drd

r drd G



















 

 

 

(30)

Latin A m erican Journal of Solids and Structures 12 (2015) 1362-1396























2

(1) 2 3 3 2 2

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

0

2 2 2 2 2 2 2

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

2 2 3 2 2

1 3 2 2 3 2 2 3 3 2

5 _{4 3 4 4 4 3}

3

32 56 64 98 6 8 3 7

3 8 10 8 4 12

b

Q h h h h h Q Q

h h Q

                                             _{   }   _    _{ }    _    _         _  _     













2 2 2 2

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

2 2 2 2 2 2 2 2 2 2

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

2 4

8 2

h h h Q Q h h h Q Q

h h Q h h Q dr

            _{ }      _  (31)



















2

(2) 2 3 3 2 3 2

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

0

2 2 2 2 2

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

2 2 2 2 2

2 1 3 1 3 2 1 2

5 _{4 3 4 12 12 21}

3

4 4 8 8 7 4 4 8 8 7 2

40 32

i b

Q h h h h h Q Q

h Q h Q h h h Q

                                          _{   }   _    _ _    _    _          _     













2 3 2 2 2 2 2 2

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

2 2 2 2 2 2 2 2 2 2 2 2

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

56 84 15

6 9 4 2 6 9 4 2

h h Q

h h Q h h Q dr

                   _{   }         _   _ _   _{ } 























2

(3) 2 3 3 2 2

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

0

2 2 2 2 2 2 2

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

2 2 3 2 2

3 1 2 1 2 2 1 2 1 2

5 _{4 3 4 4 4 3}

3

32 56 64 98 6 8 3 7

3 8 10 8 4 12

b

Q h h h h h Q Q

h h Q

                                             _{   }   _    _{ }    _    _         _  _     













2 2 2 2

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

2 2 2 2 2 2 2 2 2 2

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

2 4

8 2

h h h Q Q h h h Q Q

h h Q h h Q dr

                     where ( ) ( )2 1 i i i i i h E v



  .

Latin A m erican Journal of Solids and S tructures 12 (2015) 1362-1396



























(1) 3

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

3

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

3 2 2 2 2 3 2 2

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

2 (

2 1

1 2 3 2 1 3 3 2 2

3 1 _{8 2}

4 3 4

6 6 4 2

8

8

z h h h h Q h h Q h h Q

h h h h h h h h h

h h h Q h Q h h h Q h h h h Q h Q h h Q

h

h h h Q h h

                                     _      _ 









1)2 2 2

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

3 (1)3

2 2 2 1

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

8 3 4

2

4 3 6 12

24 3 nt

h h h Q h Q h h Q h h Q

h

h h h Q h Q h h Q q

           _{    }  _{ }          _     





















(2) 2 2 2

2 2 2 1 3 1 1 3 3 2

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

3 2 2 4 2 3 2 2 2

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

2 2 2 2 2

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

3 1 ₆

4 3 4

2 8 4

12 8

z h Q h h h h

h h h h h h h h h

h h Q h Q h h Q h Q h h h Q h h h h Q

h h Q h Q h Q h h h h

                         _{ }                























(1) 1 2 (2)2

2 3 2

3 2 2 1 3 3 1

2 2 2 2 2 2

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

3 (2)3

2 2 (1)

1 2 3 2 2

2

8

8 2

12 8 24

32

24 3 z h

h

h h Q h Q

h h Q h Q Q h h h h h h Q h Q

h

h h h Q

             _{ }         _{ } _{  }  _{ }  _        _        (32)



























(3) 3

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

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

2 3 2 2 2 3 2 2

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

2 (

2 3

1 2 3 2 3 1 1 2 2

3 1 _{8 2}

4 3 4

6 6 4 2

8

8

z h h h h Q h h Q h h Q

h h h h h h h h h

h h h Q h Q h h h Q h h h h Q h h Q h Q

h

h h h Q h h

                                     _      _ 











3)2

2 2 2

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

3 (3)3

2 2 3

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

4 3 6 12

2

8 3 4

24 3 nb

h h h Q h Q h h Q

h

h h h Q h Q h h Q h h Q q

           _{ }     _{ }          _  _    and: r Q Q Q i r i i  , 

~

Latin A m erican Journal of Solids and Structures 12 (2015) 1362-1396 4 GOVERNING EQUATIONS OF FREE VIBRATION OF THE FUNCTIONALLY

GRADED CIRCULAR SANDWICH PLATE

The governing equations of the free vibration of the sandwich plate may be derived by either using the minimum total potential energy principle or Hamilton’s principle. Based on the first approach, one may write:

0











U



K



(34)

where



,



U, and



Kare increments of the total potential energy, strain energy, and kinetic energy (energy of the inertial loads), respectively:

,

)

(

2

1













V

rz rz r

r V

T

_dV

_dV

U



















 











V

dV

w

w

u

u

K













(35)

Employing the minimum total potential energy principle taking into account Eqs. (9,11,12), leads to the following five coupled governing equations of motion of the circular sandwich plate in the cylindrical coordinate system (r, θ, z):









0

(1) (2) (3) 2 (1) 2 (1) 2 (1) 2 (1) (2) 2 (3) 2 (2) 0

(3) 2 (3) 2 (3) (1) (2) (3) (1) 2 (1) (1)

0 0 0 0 1 0

(1) (3) (2) (2) (3 2

0 0 1 1

0:

( )

2 2 2

2 2

2

u

h h h

A A A u B A A B A

h h

B A I I I u I I

h

I I I I















   

   _  _ _   _

   

   

_  _    _  _

   

 

_   _ 

 

) 2 (3) (3) 0

2

h

I



 _ 

 

 

(36-a)

(1)

2

(1) 2 (1) 2 2 (1) (1) (1) 2 (1) 2 (1) 2 (1) 2 (2) (1) (1)

0 2

(1) (1) 2 (1) 2 (1) 2 (1) (2) (1) (1) 2

1 0 0 1 0 2 2 1

0 :

2

2

2

2

2

2

2

2

h

h

h

h

B

A

u

A

h B

D

B

A

A

h

h

h

h

w

I

I

u

I

I

I

h I

r





















_



_

_

_





_

_

_

_

_



_



_

_



















_







_



















_



_

_

_

_

_

_

_



_

 









 







2

(1) (1) 0

I





_





_

_











Latin A m erican Journal of Solids and S tructures 12 (2015) 1362-1396 (2)

2

(1) (3) (2) 2 (1) (1) 2 (1) (1) (3) (2)

2 2 2 2

0

2 (2) 2 (3) 2 (3) 2 (3) (2) (2) (2) 2 (1) (2) 2 (3)

0 1 0 0

2

0 :

( ) ( )

2 2 2 2

2 2 2 2

2

h h h h

A A B u B A A A D

h h w h h

B A A I I I u

r h















 

 _{ } _{ }  _ _{ }   _{ }

 

     

    _  _



     

  _  _  _  __   _



 

   

 (1) 2 (1) (1) 2 2 (1) (3) (2) (2) 2 (3) 2 (3) (3)

1 ₂ 0 ₂ ( 0 0 ) 2 ₂ 1 ₂ 0

h h h h

I I



 I I I 



I I



 _  _   _{  }  _ 

 

     

  _  _  

(36-c)

(3)

(3) 2 (3) 2 2 2 (3) (3) 2 (2) (3) 2 (3) 2 (3) 2 (3) 0

2

(3) (3) (3) (3) 3 (3) (3) (3) 2 (3) (3)

1 0 0 2 2 1 0

(3) 2 2

0

0 :

2

2 2 2 2 2

2 2

2 2

h h h h h

B A u A B D B A

h h

w

A I I u I h I I

r h h

I















 _ _{ }  _ _{ } _{ } _

     

     

 

    

 

 _  __  _   _{ } 



    _   _

 (3) (2)

1

I



 _ 

 

 

(36-d)



(1) (1) (2) (2) (3) (3)



2 (1) (1) (1) (1) 2

(2) (2) (3) (3)

(2) (2) (3) (3) (1) (2) (3)

0 0 0

0 :

1

( )

w

w w

A A A A

r r r r

r

A A I I I w

r r r r





























    

  _  _ _  _

 



   

   

 _  _ _  _  

 

   

Latin A m erican Journal of Solids and Structures 12 (2015) 1362-1396 The resulting essential and natural boundary conditions are:









0

(1)

(1) (2) (3) 0 (1) (2) (3) 1 (1) (1) (1) 2 (1)

0

(2) (3)

(1) 2 (1) (2) 2 (3) (3) 2 (3) (3) (3) 2 (3)

(1 2 0 or 1 1 2 2 1

2 2 2 2

1 2 u

u h h

A A A A A A u A B B A

r r r r

h h h h

A B A B A B A

r r r

h A r                  _  _  _  _               _   _  _  _ _  _        

 ) (2) 2 (3) (2) ₀

2 h

B A 

 _{ }  _     (37-a) (1) (1)

(1) (1) (1) (2) (1) (1) (1) (1)

2 2 2 2 2

0

(1) (2) (1) (1) (1)

2 2 2

0

0 or

1

2 2 2 2 2

1 ₀

2 2 2

h h h h h

A B u D B B A

r r r

h h h

u D B

r















   _{ }   _{ }  _{ }  _  _   _     _            _{ } _  _  _         (37-b) (2) (1)

(1) (1) (2) (1) (1) (2) (1)

2 2 2 2 2 2 2

0 0

(2)

(1) (1) (2) 0 (2) (2) (3) (3) (2)

2 2 2 2

0 0

(2) (2) 2 (3)

0 or

2 2 2 2 2 2 2

1 1

2 2 2 2

1 1

2

h h h h h h h

A u A u B

r r r

u

h h h h

B B B u D A u

r r r r r

h

D A u

r r























   _{ } _  _{ } _                      _   _     

  2 (3) 2 (2) 2 (3) (3) 2 (3) (3)

0 _{2 2 2 2} 1 0

h h h h

B B r r











 _{ } _{  }   _   (37-c) (3)

(3) 2 (3) 2 (3) 2 (2) (3) 2 (3) (3) (3) 2 (3)

0, , , ,

(3) (2) (3) (3) (3)

2 2 2

0

0 or

1

2 2 2 2 2

1 ₀

2 2 2

r r r r

h h h h h

B A u D B B A

r

h h h

u D B

r















  _  _{ }  _{ }  _  _                 _{ } _  _  _         (37-d)



(1) (1) (2) (2) (3) (3)



(1) (1) (1) (2) (2) (2) (3) (3) (3)

0 or

,r 0

w

A A A w A A A

        



Latin A m erican Journal of Solids and S tructures 12 (2015) 1362-1396 where

( ) ( )

2 ( ) 2 ( ) ( ) 2 ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )2 ( )2 ( )

( ) ( )2 ( ) ( )2

2 2 2

( ) ( )

1 1

, , 1,2,3

1 1 2(1 )

i i i

i i i

i h i h h

i i i i

i i i i i i i i

i i i

h h h

i i i i

i i

j

A A

E v E E

B d B d A d i

v v v

D D

I

    

 

 

  

  _{ }   _{ }

  _{ }   _{ }

 _  _{  }

  _     _   _

       

   

   

   









2

( ) ( )

2

; 1,2,3; 0,1,2

i

i

h

i j i

h

d i j



 



(38)

5 DEVELOPMENT OF TAYLOR’S TRANSFORM ANALYTICAL SOLUTION

The analytical solution is developed based on Taylor’s transform method. The displacement parameters, as analytical functions, may be expressed in terms of Taylor’s series around r=0, based on a Kantorovich-type separation of variables.

(1) (1) (2) (2) (3) (3)

0

0 0 0 0

0

,

,

,

,

N N N N

k i t k i t k i t k i t

k k k k

k k k k

N

k i t k k

u

U r e

r e

r e

r e

w

W r e

   









   





























(39)

By substituting Eq. (39) into the governing Eq. (36) and performing some manipulations, the transformed form of Eq. (36) may be written as:









(1) (3) (2) (1) 2 (1) (1)

2 2

0

(1) (2) (3) (2) (3) (3) (3)

2 2 2

2 2

(1) (2) (3) 2 (1) 2 (1) 2 (

0 0 0 1 0

( 1)( 3) ( 1)( 3)

2

( 1)( 3) ( 1)( 3)

2 2 2

2

N

k k

k

k k

k k

h

A A A k k U B A k k

h h h

A B A k k B A k k

h

I I I



U I I



 



 

 _{    } _  _{  }

 _ _



   

_   _    _  _   

   

 

   _  _ 

 







1) (3) 2 (3) 2 (3)

1 0

(1) (3) (2) 2 (2) 2

0 0 1

2

0 2

k

k k

h

I I

h

I I I r





 

_  _ 

 



 

_   _  _ 

  

Latin A m erican Journal of Solids and Structures 12 (2015) 1362-1396 2

(1) 2 (1) 2 (1) (1) (1) (1)

2 2 2

0

(1) (1) (2) (1) (1) (1) (1) (1)

2 2

2 1 2 2 1

2

(1) 2 2

0

( 3)( 1) ( 3)( 1)

2 2

( 3)( 1) ( 1)

2 2

2

N

k k

k

k k k

h h

B A k k U A h B D k k

h h

B A k k A L W I h I

h I





         _  _{  }  _{     } _ _ _ _         _{ }   _  _     _   __        _{ }    _



(1) 2 (1) (1) 2 2 (1) 2 (1) 2 (2)

0 1 1 0 0

2 2 2

k

k k k

h h h

I I



U I I



r

   

  _  _  _  _  _ 

    

(40-b)

(1) (3) (2) (1) (1) (1)

2 2 2

2 2

0

2

(1) (3) (2) (2) (3) (3) (3)

2 2 2

2 2

(2) (2) (2)

( ) ( 3)( 1) ( 3)( 1)

2 2 2

( ) ( 3)( 1) ( 3)( 1)

2 2 2

( 1) N k k k k k k k

h h h

A A B k k U B A k k

h h h

A A D k k B A k k

A k W

        _{ }  _{  }  _  _{  } _ _ _ _  _{ }  _{ } _{ }        _  _               



(1) (2) (3) 2 (1) (1) 2 (1)

2 2 2 2

1 0 1 0 1 0

2

(1) (3) (2) 2 (2) (3) (3) 2 (3)

2 2 2

0 0 2 1 0

2 2 2 2

( ) 0

2 2 2

k k

k

k k

h h h h

I I I U I I

h h h

I I I I I r

          _   _  _  _   _{    }  _{ }  _{  } _ _      _  _          (40-c)

(3) 2 (3) 2 2 (3) (3) (2)

2 2

0

(3) (3) (3) (3) 2 (3)

1 2

(3) 2 (3) 2 (3) (3) (3) 2

2 1

( 3)( 1) ( 3)( 1)

2 2 2

( 1) ( 3)( 1)

2

2 ( 3)( 1)

2 2 2

N

L k

k

k k k

k

h h h

B A k k U A B k k

h

A k W D B k k

h h h

D B A k k I



       _  _{  }  _  _{  } _ _ _ _       _   _  _         _  _  _         



(3) 2 0 2

(3) (3) 2 (2) (3) (3) (3) 2 (3)

2 2 2

0 1 2 2 1 0 0

2 2 2

k

k

k k

I U

h h h

I I I h I I r







         _                         (40-d)



(1) (1) (2) (2) (3) (3)



2 (1) (1) (1) (2) (2) (2)

2 1 1

0

(3) (3) (3) (1) (2) (3) 2

1 0 0 0

( 2) ( 2) ( 2)

( 2) ( ) 0

N

k k k

k

k

k k

A A A k W A k A k

A k I I I W r

Latin A m erican Journal of Solids and S tructures 12 (2015) 1362-1396

By solving Eq. (40), an algebraic homogeneous system of equations including the unknown displacement parametersUk_2,(1)k₂, (2)k₂, k(3)₂and WL2(k=0,1,2,…) is obtained. The essen-tial (geometric) boundary conditions must be incorporated to make this system of equations a deterministic one. Moreover, the regularity conditions at the center of the axisymmetric plate have to be imposed:

0 0 (1) (1) 0 0 (2) (2) 0 0 (3) (3) 0 0 0 1 0 0 0 0 0 0 0 0 0 0 r r r r r r u U Q W







                      _{   }   _{  }  (41)

By substituting Eq. (39) into the boundary conditions Eq. (37) the transformed form of Eq. (37) becomes:

























2 0 1

(1) (2) (3) (1) (2) (3) (1) 2 (1) (1)

1 1

0

(1) (2) (3) (2) (1) (2) (3) (2)

2 2 2 2

1 (3) 2 (3)

0,

1

1 1

2

1 ₁

2 2 2 2

1 2 N k k L N

k k k

k

k k

k

u U r

or

h

A A A k U A A A U B A k

r

h h h h

A B A A B A k

r

h

B A k

          _{      } _  _{ }  _ _       _   _ _   _         _  _    





(3) (3) 2 (3) (3) (1) 2 (1) (1)

1 1 ₂ k 1 ₂ k k 0

h h

B A B A r

r r                 (42-a)









2 (1) (1) 0 1

(1) (1) (1) (2) (1) (1) (1)

2 2 2 2

1 1 1 1

0

(1) 2 (1) 2 (1) 2 (2) (1) 2 (1) (1)

0,

or

1 1

2 2 2 2

1 1

2 2 2 2

N k k r b k N

k k k L

k

k k k L

r

h h h h

A B k U D B k

h h h h

B A U D B

Latin A m erican Journal of Solids and Structures 12 (2015) 1362-1396

















2 (2) (2) 0 1

(1) (1) (2) (1) (1) (2)

2 2 2 2 2 2

1 1 1

0

(1) (1) (1) (1) (2) (2) (2) (2)

2 2

1 1 1

(2) (

0,

or

1 1

2 2 2 2 2 2

1 1

1 1 1

2 2 1 N k k r b L N

k k k k k k

k

k k k k k

k

r

h h h h h h

A k U A U

r

h h

B L B B k U B U D k

r r D r



               _  _{   } _  _{   }   _ _ _ _              













2) 2 (3) 2 (3) 2 (2) 2 (3) (3)

1 1 1 1

(3) (3) (2) (3) (3)

2 2 2 2

1 1

2 2 2 2

1 1 ₀

2 2 2 2

k k k k

k

k k k k

h h h h

A k U B k

h h h h

A U B r

r r         _     _         _     _     (42-c)









2 (3) (3) 0 1

(3) 2 (3) 2 (3) 2 (2) (3) 2 (3) (3)

1 1 1 1

0

(3) 2 (3) 2 (3) 2 (2) (3) 2 (3) (3)

0,

or

1 1

2 2 2 2

1 1

2 2 2 2

N k k r b L N

k k k k

k

k k k k

r

h h h h

B A k U D B k

h h h h

B A U D B

r r



_             _  _  _{   }  _{ }  _{ } _ _ _  _{ } _         _  _{ }     _{ }  _        





0 k r   (42-d)









1 0 2

(1) (1) (2) (2) (3) (3) (1) (1) (1) (2) (2) (2) 1

0

(3) (3) (3)

0, or 1 0 N k k r b k N

k k k

k

k k

w W r

A A A k W A A