Three dimensional mechanical behaviors of in-plane functionally graded plates

Three dimensional mechanical behaviors of in-plane functionally graded plates

Journal Pre-proofs Three dimensional mechanical behaviors of in-plane functionally graded plates Pengchong Zhang, Chengzhi Qi, Hongyuan Fang, Wei He P...

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Journal Pre-proofs Three dimensional mechanical behaviors of in-plane functionally graded plates Pengchong Zhang, Chengzhi Qi, Hongyuan Fang, Wei He PII: DOI: Reference:

S0263-8223(19)34447-2 https://doi.org/10.1016/j.compstruct.2020.112124 COST 112124

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Composite Structures

Received Date: Revised Date: Accepted Date:

22 November 2019 17 February 2020 22 February 2020

Please cite this article as: Zhang, P., Qi, C., Fang, H., He, W., Three dimensional mechanical behaviors of inplane functionally graded plates, Composite Structures (2020), doi: https://doi.org/10.1016/j.compstruct. 2020.112124

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1

Three dimensional mechanical behaviors of in-plane functionally

2

graded plates

3

Pengchong Zhanga,b*, Chengzhi Qi a,b, Hongyuan Fangc, Wei Hea

4 5 6 7 8 9 10 11 12

(a. School of Civil and Transportation Engineering, Beijing University of Civil Engineering and Architecture, Beijing 102616, China; b. Beijing Advanced Innovation Center for Future Urban Design, Beijing University of Civil Engineering and Architecture, Beijing 100044, China; c. College of Water Conservancy & Environmental Engineering, Zhengzhou University. Zhengzhou, 450001, China; Corresponding

author:

Pengchong

Zhang,

[email protected],

+86-15382171875

Abstract

13

A semi-analytical solution procedure to investigate the distributions of

14

displacement and stress components in the in-plane functionally graded plates based

15

on the scaled boundary finite element method (SBFEM) in association with the

16

precise integration algorithm (PIA) is developed in this paper. The proposed approach

17

is applicable to conduct the flexural analysis on functionally graded plates with

18

various geometric configurations, boundary conditions, aspect ratios and gradient

19

functions. The elastic material parameters of functionally graded plates discussed here

20

are mathematically formulated as power law, exponential and trigonometric functions

21

varied along with the in-plane directions in a continuous pattern. Only a surface of the

22

plate parallel to the middle plane is required to be discretized with two dimensional

23

high order spectral elements, which contributes to reducing the computational expense.

24

By virtue of the scaled boundary coordinates, the virtual work principle and the

25

internal nodal force vector, the basic equations of elasticity are converted into a first

26

order ordinary differential SBFEM matrix equation. The general solution of the

27

governing equation is analytically expressed as a matrix exponential with respect to

28

the transverse coordinate z. According to the PIA, the stiffness matrix from the matrix

29

exponential can be acquired. Considering that the PIA is a highly accurate method,

30

any desired accuracy of the displacement and stress field can be obtained. The entire

31

derivation process is built on the three dimensional elasticity equations without 1

1

importing any assumptions on the plate kinematics. Comparisons with numerical

2

solutions available from prevenient researchers are made to validate the high accuracy,

3

efficiency and serviceability of the employed technique. Additionally, circular and

4

perforated examples are provided to highlight the performance of the developed

5

methodology and depict the influences of boundary conditions, thickness-to-length

6

ratios and gradient indexes on the deformable behaviors of in-plane functionally

7

graded plates.

8

Keywords: Functionally graded plates; In-plane inhomogeneity; Mechanical

9

behaviors; Scaled boundary finite element method; The precise integration algorithm

10

1. Introduction

11

As a kind of advanced composite materials, the functionally graded materials

12

firstly introduced by Japanese scientists [1] are characterized by the material

13

properties transiting smoothly and continuously along with one desired or more

14

spatial coordinates to avoid the problem of the stress concentration, sharp interfaces in

15

laminated composite structures and the existence of residual stresses, delamination

16

and cracks. Recently, the functionally graded materials are extensively utilized in

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plate structures, coined as functionally graded plates, to improve their mechanical

18

properties. Since plates are usually seen as an important structural component, it is

19

necessary to carry out an investigation on the variations of displacement and stress

20

fields in functionally graded plates to confirm the safety and better serve the

21

engineering applications. To the best of authors’ knowledge, the bending responses of

22

functionally graded plates with material parameters formulated as functions of the

23

transverse coordinate attract most researchers’ attention. However, studies reveal that

24

functionally graded plates with material properties gradation along the in-plane

25

directions are practical occasions [2] and few explorations are conducted on the

26

flexural analysis of these structures. Therefore, the present paper will pay attention to

27

the mechanical behaviors of in-plane functionally graded plates based on a

28

semi-analytical numerical method.

29

Owing to the widespread applications of functionally graded plates in 2

1

engineering structures, numerous scientists and engineers develop various theories

2

and implement many analytical and numerical techniques to predict the deformable

3

characteristics of them. Comprehensive literature reviews related to the structural

4

responses of functionally graded plates are presented by Ref. [3-7]. With the aid of the

5

Plevako’s solution [8], Kashtalyan [9] carried out the three dimensional static analysis

6

of simply supported functionally graded material plates subjected to sinusoidal loads.

7

By means of the Fourier series expansion approach and the classical plate theory, Chi

8

and Chung [10-11] made effort on solving the bending problem of simply supported

9

functionally graded rectangular plates. The Poisson’s ratio of the plate is a constant

10

but the Young’s moduli is expressed as power-law, sigmoid and exponential functions.

11

Talha and Singh [12] introduced a new finite element model based on the high order

12

shear deformation plate theory to investigate the bending and free vibration responses

13

of functionally graded plates. Singha et al. [13] combined the first order shear

14

deformation plate theory and the finite element method to develop a high precision

15

plate element for analyzing the bending behaviors of functionally graded square plates

16

under simply supported and clamped boundary conditions. A three dimensional elastic

17

solution to explore the static flexure of transversely isotropic functionally graded

18

material plates with the simply supported boundary constraint was obtained by

19

Woodward and Kashtalyan [14]. Aghdam et al. [15] exploited the extended

20

Kantorovich method and the first order shear deformation plate theory to predict the

21

static responses of functionally graded material sector plates with the clamped

22

boundary condition. Jha et al. [16] applied a higher order shear and normal plate

23

theory to conduct a study on the static and dynamic responses of simply supported

24

functionally graded rectangular plates with the material properties varied as power

25

law and exponential formulations through the thickness direction. Based on the

26

non-polynomial high order shear deformation plate theory, Mantari and Soares [17]

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introduced a new kind of four-nodded quadrilateral isoparametric element to evaluate

28

the displacements and stresses in functionally graded material plates. The

29

isogeometric analysis based on the non-uniform rational B-spline basis functions

30

combined with first-order or high order shear deformation plate theory were proposed 3

1

by many researchers [18-24] as useful tools to study the static flexure, free vibration

2

and buckling responses of functionally graded material plates with various shapes.

3

Thai and Choi [25] made utilization of an improved refined plate theory to exhibit the

4

static and dynamic responses of functionally graded rectangular plates under the

5

simply supported boundary condition. Akavci and Tanrikulu [26] proposed a new

6

quasi-three dimensional higher order shear deformation plate theory to implement the

7

static and dynamic analysis of thick functionally graded plates. Nguyen et al. [27]

8

introduced a novel high order hyperbolic shear deformation plate theory to conduct

9

the static flexure, free vibration and buckling analysis of functionally graded plates

10

with the material coefficients formulated as the power-law function. Pradhan and

11

Chakraverty [28] exploited the classical plate deformation theory to present the

12

bending analysis of functionally graded rectangular plates under clamped, simply

13

supported and free boundary conditions. As an extension of the displacement function

14

method, Lu et al. [29] released the three-dimensional elasticity solutions of simply

15

supported and clamped transversely isotropic functionally graded material circular

16

plates under the action of axisymmetric loads. Fallah and Khakbaz [30] took

17

advantage of the first order shear deformation theory and the extended Kantorovich

18

approach to present the static analysis of solid and annular sector functionally graded

19

plates subjected to external transverse forces. By virtue of the meshfree moving

20

Kriging approach in conjunction with the first-order shear deformation theory, Vu et

21

al. [31] examined the flexural and free vibration responses of functionally graded

22

material plates. Gupta and Talha [32] applied the non-polynomial higher-order shear

23

and normal deformation theory to present the static and dynamic behaviors of porous

24

functionally graded square plates with material properties gradation as a power law

25

function. LomtePatil et al. [33] obtained the three dimensional elastic formulations of

26

functionally graded plates under the simply supported boundary condition to examine

27

the distributions of displacement and stress fields. Based on the generalized England’s

28

theory, Yang et al. [34] developed three dimensional solutions to the bending problem

29

of transversely isotropic functionally graded elliptical plates. Kumar et al. [35]

30

introduced new trigonometric and algebraic high order shear deformation plate 4

1

theories incorporated with the meshfree approach to display the bending behaviors of

2

simply supported functionally graded plates. Merdaci and Belghoul [36] employed a

3

simple high order shear and normal plate theory to explore the variations of

4

displacement and stress components in porous functionally graded plates.

5

From the literature review listed above, it is clear that the static flexure features

6

of functionally graded plates with the material stiffness varied through the thickness

7

direction have been extensively examined by many plate theories and numerical

8

approaches, like the inelasticity separated finite element method [37-40]. However,

9

few papers discuss the bending behaviors of functionally graded plates with

10

macroscopic material properties grading along the in-plane directions, which will

11

produce great importance on the distributions of structural components. Zenkour and

12

Fares [41] employed the third order shear deformation theory and the small parameter

13

method to analyze the variations of central deflections and stresses in three and four

14

layered functionally graded square plates with the in-plane material inhomogeneity.

15

With the aid of the state space method, Zenkour et al. [42] investigated the

16

distributions of transverse deformations and stresses in variable-thickness in-plane

17

functionally graded rectangular plates under various boundary conditions. Fereidoon

18

et al. [43] made use of the extended Kantorovich method and the classical plate theory

19

to conduct a study on the bending analysis of radially functionally graded thin sector

20

plates. By virtue of the multi-term extended Kantorovich method and the first order

21

shear deformation plate theory, Mousavi and Tahani [44] developed an analytical

22

method to solve the static responses of functionally graded sector plates with material

23

parameters changed along the radial direction. Rad [45] utilized the state space

24

method combined with the differential quadrature approach to make a research on the

25

static responses of functionally graded circular plate with through-thickness and radial

26

inhomogeneity resting on the Winkler-Pasternak foundation. Built upon the technique

27

of the state space and the differential quadrature, Rad and Shariyat [46] acquired an

28

elastic solution on the interaction between the non-uniform Winkler-Pasternak

29

foundations and functionally graded annular plates with material coefficients grading

30

along the radial and thickness directions. Shariyat and Mohammadjani [47] put 5

1

forward a new finite element model to discuss the bending behaviors of rotating

2

functionally graded circular and annular plates embedded on the non-uniform elastic

3

Winkler foundation. What’s more, variations of the Young’s modulus, Poisson’s ratio

4

and density obey the formulations of the transverse and radial coordinates. A

5

Levy-type solution was proposed by Yu et al. [48] to study the distributions of

6

deflections and bending moments in thin rectangular plates with in-plane

7

non-homogeneous stiffness. Amirpour et al. [49] gave out analytical solutions and the

8

numerical modeling based on the finite element method to simulate the changing

9

patterns of transverse displacements and stresses through the length and thickness

10

directions in thin functionally graded rectangular plates with the material

11

heterogeneity along the length. Kumari et al. [50] took advantage of the Reissner-type

12

mixed variational principle combined with the mixed-field multi-term extended

13

Kantorovich method to investigate the distributions of displacements and stresses

14

along the longitudinal and thickness directions for the in-plane functionally graded

15

rectangular plates. Variations of the in-plane thermal stress and strain fields in

16

functionally graded circular and annular plates with the gradation through the radial

17

and tangential directions were examined by Demirbas and Apalak [51]. Lieu et al. [52]

18

carried out the isogeometric analysis on the static and dynamic responses of

19

variable-thickness functionally graded plates. The variations of material constants are

20

expressed as power law functions and the thickness of the plate is a function of the

21

in-plane x and y coordinates. Singh et al. [53] utilized the extended Kantorovich

22

method to explore the flexural behaviors of axially functionally graded angle-ply

23

panels under the cylindrical bending with various boundary conditions. A finite

24

element graded model built upon the commercial software ABAQUS and a set of

25

original experimental devices were introduced by Amirpour et al. [54] to estimate the

26

change rules of deflections in polymeric functionally graded plates. With the help of

27

the hyperbolic shear deformation plate theory and the isogeometric analysis method,

28

Farzam and Hassani [55] carried out a research on the static, free vibration and

29

buckling responses of functionally graded rectangular plates with material constants

30

changed along the transverse and in-plane directions. Aided by the Kantorovich and 6

1

power series methods, Ravindran and Bhaskar [56] evaluated the variations of

2

displacement and stress components in unidirectionally functionally graded plates

3

with in-plane power-law non-homogeneity.

4

The present paper introduces a semi-analytical technique to investigate the

5

bending behaviors of in-plane functionally graded plates with various geometric

6

shapes, boundary conditions, thickness-to-length ratios and gradient functions based

7

on the SBFEM and PIA for the first time. Compared with the traditional finite element

8

method (FEM) and boundary element method (BEM), the SBFEM developed by Song

9

and Wolf [57-58] only requires to discretize the boundary of the research domain and

10

the fundamental solutions are not essential. Moreover, the basic unknowns of the

11

research issue can be formulated analytically along the radial direction.

12

Correspondingly, solutions in the circumferential direction can be obtained by virtue

13

of the interpolation functions similar as the FEM. As a bright and promising

14

semi-analytical method, the SBFEM has been successfully utilized to solve the

15

problem of the interaction between the soil and structures [59-60], the crack

16

propagation [61-62], the non-linear analysis [63-65], the boundary value problems

17

[66-67], the seepage analysis [68] and so on. Recently, researchers have applied the

18

SBFEM to analyze the static and dynamic responses of structural components, such as

19

beams [69], plates [70-74] and shells [75].

20

Xiang et al. [76] carried out implementation on the free vibration and buckling

21

analysis of functionally graded rectangular and skew plates with material properties

22

obeying a power-law variation according to the inclusion volume fraction along the

23

length direction. However, as far as the authors know, there has been no works

24

contributing on exploring the mechanical bending behaviors of functionally graded

25

plates with material coefficients following arbitrary functions of the in-plane

26

coordinates based on the SBFEM. Therefore, the research of this paper is conducted

27

as the first effort. Additionally, to improve the precision of the displacement and

28

stress components, the PIA [77] as a highly accurate approach is exploited and

29

calculated results can reach up to the limited precision of the computer used.

30

When the SBFEM is utilized to investigate the static flexure of in-plane 7

1

functionally graded plates, the scaled boundary coordinates, the virtual work principle

2

and the internal nodal forces are employed to convert the governing partial differential

3

equations into a simple first order ordinary differential matrix equation whose general

4

solution is a matrix exponential. As a version of 2N method, the PIA is convenient and

5

accurate to solve the matrix exponential. Then, the displacements and stresses of the

6

plates can be evaluated. This paper is organized in the following. Section 2 gives out

7

the theoretical derivation process of the governing matrix equations based on the

8

SBFEM. Section 3 provides the solution procedure of displacements and stresses by

9

virtue of the PIA. The accuracy and efficiency of the proposed technique is validated

10

and two more numerical examples are demonstrated to discuss the effect of elastic

11

parameters in Section 4. Finally, the main conclusions are drawn in Section 5.

12

2. Governing equation of the in-plane functionally graded plate

13

In this section, the governing SBFEM equation of the in-plane functionally

14

graded plates with the length l, width b and uniform thickness t is formed, as

15

illustrated in Fig. 1. The theoretical derivation begins with the three dimensional basic

16

equations of elasticity without adding any assumptions of the plate kinematics.

17

What’s more, only three translational displacements are selected as the basic unknown

18

variables. When the SBFEM is applied to analyze the behaviors of plates, only a

19

surface parallel with the middle plane is required to be discretized and any type of

20

displacement-based plane elements can be utilized. Owing to the characteristics that

21

the integration points coincide with the field nodes and the curve boundary can be

22

better modelled, the two dimensional high order spectral elements are chosen to mesh

23

the functionally graded plate in this paper to improve the computational accuracy and

24

efficiency. Fig. 2 demonstrates the square plate discretized with four fourth-order

25

spectral elements.

26

It’s necessary to point out that the material coefficients of functionally graded

27

plates considered here are assumed to vary continuously and smoothly throughout the

28

in-plane directions according to arbitrary distributions which are defined by

29

mathematical expressions, such as power law, exponential, trigonometric functions 8

1

and so on. In other words, the constitutive matrix is a function of the in-plane

2

coordinates x and y while it remains a constant along the thickness direction.

3

The elastic displacement components along x, y and z directions are formulated

4

as ux=ux(x,y,z), uy=uy(x,y,z) and uz=uz(x,y,z). A three dimensional vector arranged as

5

{u}={u(x,y,z)}=[uz ux uy]T is introduced to facilitate the following derivation.

6

From the elasticity, the geometric equation can be denoted as

     zz  xx  yy  xy  yz  xz 

T

  L u 

(1)

7

in which εij (i,j=x, y, z) represents the elastic strain field and the differential operator

8

[L] is expressed as   z 0    [ L]= 0 x    0 0 

9 10

0 0  y

 y

0  y  x

0  z

12 13

15

T

(2)

plate in the Cartesian coordinate system z–x–y can be given as T

  D  x , y    

(3)

where [σzz σxx σyy τxy τyz τxz]T is the stress vector and [D(x,y)] is the elastic matrix. Meanwhile, the key equilibrium equations absence of the body forces can be written as

 xx  xy  xz   0 x y z  xy  yy  yz   0 x y z  xz  yz  zz   0 x y z 14

0

        

At the same time, the constitutive equation of the in-plane functionally graded

    zz  xx  yy  xy  yz  xz  11

 x  z

(4) (5) (6)

By means of the differential operator [L] in Eq, (2), the equilibrium equations can be simplified as

 L     0 T

(7)

16

In the SBFEM, a set of scaled boundary coordinates η and ζ is introduced to

17

transform the basic partial differential equations into a second order ordinary

18

differential matrix one. Moreover, the scaling center O at the infinity is selected. So 9

1

the transverse lines perpendicular to the middle plane are chosen as the research

2

objectives. The displacement and stress components varied along the transverse

3

direction can be expressed analytically. The coordinates (x(η,ζ), y(η,ζ)) of any point in

4

the plate can be obtained according to the interpolation function [N]=[N(η,ζ)]=[N1(η,ζ)

5

N2(η,ζ) …] x( ,  )   N  x y ( ,  )   N  y

6 7 8

(8)

in which {x} and {y} are the nodal coordinates of the high order spectral elements. With the help of the employed scaled boundary coordinates, the elastic matrix [D(x,y)] is rewritten as

 D(x, y)  D( N  x , N  y)   D(, ) 9 10

By virtue of the Jacobian matrix, the conversion from the scaled boundary coordinates to the global coordinate system is formulated as  /    / x     J ( ,  )    /    / y 

11

(10)

with the Jacobian matrix  N ,  x J(, )    N ,  x

12

 N,  y   x,   N,  y x,

y,  y, 

14

(12)

From Eq. (10), the differential operator [L] in the scaled boundary coordinate system z–η–ζ is expressed as [ L ]  [ b1 ]

15

(11)

and its determinant

J(, )  x, y,  y, x, 13

(9)

    [b 2 ]  [b 3 ] z  

(13)

where the coefficient matrices [b1], [b2] and [b3] are denoted as 1 0  0 1 [b ]   0 0  0

0 0 0 0 0 1

0  0 0  0 0 0 0   0   y, 0  0 0  y, 0   0  x,  0 0  x,  0 1 0 2   [b3 ]  1   [b ]  x,  y,  0 J 0  x, y,  J 0     1 0   x, 0   x, 0 0    y 0 0  y  0   ,   , 0 0 

(14)

16

Similar with the coordinate interpolation, the elastic displacement field {u(z,x,y)}

17

at any point in the functionally graded plate can be gained with the aid of the shape 10

1

function matrix  Ν  N   {u  z , x, y }  {u  z , ,  }    0    0

2 3

0 0  u z (z )   N  0  u x (z )   Ν u  z  0  N  u y (z )

Considering the relationship between the displacements and strains, the strain field can be rewritten as

    B 1  u  z , z   B 2  u  z  4 5

(15)

(16)

with  B 1    b1   N  and  B 2    b 2   N ,   b 3   N , Accordingly, the elastic stresses in Eq. (3) can be formulated as { }  [ D  ,  ](  B1  {u ( z )}, z   B 2  {u ( z )})

(17)

6

Aided by a series of derivations based on the virtual work principle, integration

7

by parts and corresponding boundary conditions, the resulting SBFEM governing

8

matrix equation for the in-plane functionally graded plates is denoted as

[ E 0 ]u ( z ), zz  ([ E1 ]T  [ E1 ]) u ( z ), z  [ E 2 ]u ( z )  0 9

(18)

in which [E0], [E1] and [E2] are the SBFEM coefficient matrices and expressed as [E 0 ] 

[E

1 1

1

1

1

1

1

1

1

1

  ]  

[E1]  2

1

1

 

[ B 1 ]T [ D  ,  ][ B 1 ] J d  d 

(19)

[ B 2 ]T [ D  ,  ][ B 1 ] J d  d 

(20)

[ B ] [ D  ,  ][ B ] J d  d 

(21)

2 T

2

10

It’s noteworthy to mention that only a two dimensional spectral element is taken

11

as an example in this section. A process of element-by-element assembling

12

implemented in the traditional FEM is needed to calculate the whole plate. In addition,

13

only the necessary equations are listed in the above derivation and readers can refer to

14

Man et al. [71-72] for more details.

15

3. Solutions of the displacement and stress fields

16 17

To further simplify the governing equation, a vector named the internal nodal force is employed, which is regarded as the dual vector of the displacement {u(z)}

q  z  E u  z 0

,z

18 19

 E1  u  z  T

(22)

By virtue of the introduced vector {X(z)}={{u(z)} {q(z)}}T, the SBFEM governing equation is converted into a first order ordinary differential matrix one 11

 X  z 

,z

1

   Z  X  z 

with the coefficient matrix [Z] T 1   E 0   E1   Z    2 1 0 1 1 T    E    E   E   E 

2

(23)

  1 0 1    E   E     E 0 

1

(24)

The general solution for Eq. (23) is expressed as

 X  z   e   c Z z

(25)

3

in which e-[Z]z is a matrix exponential and {c} is the integration constant vector

4

determined by the boundary constraints.

5

From Eq. (25), it is apparent that the displacements and internal nodal forces of

6

the in-plane functionally graded plate can be expressed analytically along the

7

thickness direction.

8

The Taylor series, Padé expansion and other numerical techniques have been

9

exploited to solve the matrix exponential in Eq. (25). To improve the computation

10

accuracy, the PIA is employed to calculate the matrix exponential in this paper. As a

11

version of 2N method, the PIA is convenient and accurate to compute the matrix

12

exponential and predicted results can reach up to the limited precision of the computer

13

used.

14 15

Substituting the corresponding z coordinates of the top and bottom planes, it is easy to gain the following formulae

 X B   c

 X T   e Z t c  exp    Z  t  c

(26)

16

where {XB} and {XT} stand for the vectors of displacements and internal nodal forces

17

at the bottom and top planes and t represents the thickness of the plate.

18 19

On the basis of the relation between the internal nodal forces and the boundary tractions, Eq. (26) is reformulated as  uB     uB     c1    =    q  F c        B        2   B  

  uT    uT    c1       exp    Z  t    q F c        T     T    2    

(27)

20

in which {FB} and {FT} are external forces caused by tractions on the bottom and top

21

planes.

22 23

In the PIA, the whole thickness t is firstly divided into 2N layers and each layer owns the equal thickness ς. Eq. (27) can be rewritten as 12

 XT   exp   Z  t 2N  1

c  exp   Z  

2N

c  T2 c  Tˆ c N

(28)

ˆ  T 2N . are expressed as T  exp    Z    and T

where T and Tˆ

In the above formulation, T can be calculated by the Taylor’s expansion

2

T  I     Z    3

2N

2 3 4 5 1 1 1 1   Z        Z        Z        Z     ......  2! 3! 4! 5!

(29)

with a unit matrix I .

4

It is obvious that the thickness of one layer is extremely small. So in Eq. (29) the

5

fourth order Taylor series is enough to make sure the accuracy. Then T is denoted

6

as T  I  Ta

7 8



Ta     Z    

2 3 4 1 1 1   Z       Z       Z    2 6 24

can be given out according to the successive factorization and adopting the

recursive formula N times N 1

ˆ   I  T 2   I  T 2   I  T 2 T a a a N

ˆ i 1 ˆ i 1  T ˆ i 1  T ˆ i  2T T r r r r 9 10 11

(30)

N 1

 i  1, 2,..., N 

ˆN  IT r

(31)

ˆ0 T T r a

(32)

From the above derivation, it is clear that the whole computations are built on the standard algebraic matrix operations to ensure the stability of the proposed method. On account of Eqs. (31) and (32), Eq. (27) can be rewritten as

ˆ   u   ˆ T ˆ  c   T ˆ T uT   ˆ c1  T B 1 11 12 11 12        T  =   ˆ T ˆ  c2 T ˆ T ˆ   FB  FT  c2 T  21 22  21 22 

(33)

12

Grouping the unknown displacements and the exerted boundary tractions into

13

different sides of Eq. (33), the stiffness equation for the in-plane functionally graded

14

plate can be acquired.

15

u   F  K  uB    FB   T   T 

(34)

Tˆ 12  1 Tˆ 11 ˆ ˆ 1 ˆ  T21  T22 T12 T11

(35)

with the stiffness matrix 

Κ    ˆ 16 17 18

 Tˆ 12  1   Tˆ 22 Tˆ 12  1 

According to Eqs. (25) and (34), it is of convenience to solve the displacement components at any location of the plate when the external forces are known. From the expression of the internal nodal force, the derivative of the vector {u(z)} 13

1

is defined as

u  z 

,z

2 3

  E 0 

q  z   E  u  z  1 T

(36)

By dint of Eqs. (17) and (36), the stress field of the in-plane functionally graded plate is formulated as { }  [ D  ,  ](  B1   E 0 

4

1

1

q  z    E  u  z    B {u( z)}) 1 T

2

(37)

4. Numerical examples

5

Four numerical examples related to the mechanical behaviors of square, circular

6

and perforated in-plane functionally graded plates are provided to display the

7

accuracy, applicability and feasibility of the proposed technique based on the SBFEM

8

and PIA in this section. By virtue of the first two functionally graded square plates,

9

the accuracy and serviceability of the present semi-analytical method have been

10

established by comparing with the elastic solutions of the classical plate theory and

11

other numerical models available in the existing literatures. In addition, extra circular

12

and perforated plates are offered to discuss the influence of boundary conditions,

13

gradation functions, thickness-to-length ratios and other parameters on the variations

14

of displacement and stress components in the in-plane functionally graded plates.

15

4.1 Verifications of central deflections

16

To demonstrate the high accuracy of the present approach, the deformable

17

responses of a square functionally graded plate under the action of a sinusoidal

18

pressure q=sin(πx/l)sin(πx/b) applied on the top plane are considered. The square plate

19

is discretized with four spectral elements, as plotted in Fig. 2. The plate is constituted

20

by a mixture of ceramic and metallic phase materials. The material coefficients of

21

these two constituents are Ec=348.43GPa, υc=0.24 and Em=201.04GPa, υm=0.3262

22

respectively, in which the notations of c and m symbolize the ceramic and metallic

23

materials. Young’s modulus and Poisson’s ratio are assumed to vary smoothly along

24

the in-plane directions in terms of the inclusion volume fraction denoted as power law

25

formulae (38) and (39) concerning x and y coordinates. In other words, the elastic

26

parameters for the square plate is expressed as E=EcVc+EmVm, υ=υcVc+υmVm and

27

Vc+Vm=1. There are three types of boundary conditions CSFS, SSSS and CCCC under 14

1

discussion and S, C and F represent the simply supported, clamped and free boundary

2

constraints, respectively. The length of the plate is defined as l=b=1m and three

3

different thickness-to-length ratios S=l/t=5, 50, and 100 are selected. Additionally, the

4

2 3 central deflections are normalized as uz  l 2, b 2  uz Ect l , where Ec stands for

5

the elastic modulus of the ceramic material. To depict the convergence study, the

6

CSFS plate with ky=1.0 is shown as an example and its corresponding dimensionless

7

deflections are released in Tables 1-2. It is noticed that with the increase of element

8

orders the results acquired by the employed technique converge to the numerical

9

solutions in Ref. [52]. The non-dimensional transverse displacements under different

10

material gradient indexes, boundary conditions, thickness-to-length ratios and

11

gradation functions are listed in Tables 3-8. From all tables, it is apparent that

12

achieved results given by the SBFEM and PIA are in good agreement with available

13

solutions from Lieu et al. [52], which means that the accuracy of the introduced

14

methodology to analyze the static flexure of in-plane functionally graded plates is

15

verified.

16

Model 1: k

x x  y Vc  x , y       l  b

17

ky

(38)

Model 2: Vc  x, y   Vc , x  x  Vc , y  y    2x  l 0 x    2   l  Vc , x  x    kx 2x  l   xl  2  l  2  kx

18

  2 y k y b 0 y    2   b  Vc , y  y    ky 2y  b   yb  2  b  2 

(39)

4.2 Validations of displacements and stresses

19

To further exhibit the effectiveness of the employed technique, an orthotropic

20

square functionally graded plate with material coefficient variations according to the

21

volume fraction distribution expressed as a quadratic power law function along the x

22

direction, i.e. Vf= Vf0+4(Vfmax-Vf0)(x/l-x2/l2), is taken into account. Similar with the

23

foregoing example, the plate is subjected to a double sinusoidal normal traction 15

1

q(x,y)=q0sin(πx/l)sin(πy/b) with the amplitude q0=1N/m2 on the top plane. Other

2

geometrical parameters of the plate are set as l=b=1m, S=l/t=5, 10, 20, 50 and 100.

3

The simply supported boundary constraint is prescribed at four lateral surfaces:

4

uz=ux=0 at y=0 and y=b, uz=uy=0 at x=0 and x=l. To facilitate the following analysis,

5

the non-dimensionalized displacement and stress components are defined as

6

u

7

100 E m 100 100 u x , u y  ,  xz ,  yz    xz ,  yz  ,  xx ,  yy ,  xy   2  xx ,  yy ,  xy  ,   3 S tS S 100 E m and  zz   zz , in which Em represents the Young’s modulus of the uz  uz tS 4 x

,uy  

8

matrix and is equal to 3.5GPa. Tables 9-10 present the elastic transverse

9

displacements and normal stresses for the 0 and 90 plates. It can be found that

10

increasing the element orders leads the convergent results to the reference solutions

11

given by Ravindran and Bhaskar [56]. The outcomes of the displacement and stress

12

field calculated by the introduced approach under different aspect ratios and fiber

13

orientations are demonstrated in Tables 11-18 and Figs. 3-4. As observed from these

14

tables and figures, it is clear that results of displacements and stresses numerically

15

predicted by the employed methodology are in exact coincidence with those

16

three-dimensional elasticity solutions from the Ref. [56]. Therefore aided by the two

17

comparative numerical examples, the high accuracy, applicability and reliability of

18

the semi-analytical approach to investigate the mechanical behaviors of in-plane

19

functionally graded plates have been established.

20

4.3 The static analysis of a circular plate

21

Due to the high load-bearing capacity and design flexibility, circular plates are

22

extensively applied in a large number of engineering practices, such as nuclear power

23

structures, bridge decks and so on. In view of their versatile usage and few research

24

involving them, this sub-section contributes to implementing the flexural analysis of

25

circular functionally graded plates with material properties following an exponential

26

function of radial variation P(r)=P0eγr (0
27

formulation means that both the elastic modulus and Poisson's ratio vary along the

28

radial direction and P0 represents the material coefficients at the center of the plate.

29

The material constants at R=0 are selected as E0=206GPa and υ0=0.3. The circular 16

1

plate under the action of a uniform transverse load q=1N/m2 on the top plane with the

2

clamped boundary condition is taken into account. In other words, the elastic

3

displacements around the edge of the circular plate are defined as uz=ux=uy=0. It is

4

necessary to point out that the same normalized forms as the above example are

5

adopted and variables of uz located at (0, 0, z), ux at (R/2, 0, z), σxx at (0, 0, z) and τxy at

6



7

three different aspect-ratios S=l/t=10, 50 and 100 are under consideration, in which l

8

is defined as the characteristic length l=2R. Distributions of displacement and stress

9

components along the thickness of the circular plate with the gradient parameter γ=1

10

are illustrated in Figs. 6-9. From all figures, it is obvious that with increasing the

11

thickness-to-length ratios of the circular plate, the amplitudes of dimensionless

12

displacements and stresses decrease. In Figs. 6-7, it can be seen that the elastic

13

displacement curves are nearly straight lines. Additionally, the deflections almost

14

remain constants along the thickness but the maxima of the displacement ux and

15

normal stress σxx occur at the bottom and top planes. As for the shear stress τxy, the

16

distribution curve for S=10 is quite different from those for S=50 and 100. It can be

17

indicated that thickness-to-length ratios play an important role in determining the

18

mechanical behaviors of circular in-plane functionally graded plates.

19

4.4 The bending responses of a perforated plate

2 R 4, 2 R 4, z



attract the authors’ attention. The radius of the plate R=1m and

20

In many practical projects, plates are needed to open holes to satisfy some design

21

requirements, provide access convenience for electrical lines, cables and fuel pipes, or

22

only reduce the weight of structures. Cutouts could impose a significant impact on the

23

mechanical responses of plates. To further depict the feasibility and reliability of the

24

proposed approach, the deformable behaviors of a circular in-plane functionally

25

graded plate with a circular hole are investigated in this sub-section. Fig. 10

26

demonstrates the plane view of the perforated plate with the outer radius Ro=1m and

27

the inner one Ri=0.4Ro. Meanwhile, the perforated circular plate is discretized with

28

four spectral elements. The characteristic length l=2Ro and radius Rc=(Ro+Ri)/2 are

29

introduced. The uniform pressures q=1N/m2 are distributed over the area from Rc to Ro. 17

1

The mechanical responses of the perforated plate at point A (Ri, 0, 0) under four

2

thickness-to-length ratios S=l/t=10, 20, 50 and 100 are examined. Simply supported

3

and clamped boundary conditions are imposed on the outer circular edge while the

4

inner one is free. The simply supported constraint is set as uz=0. The same material

5

parameters as those in Section 4.3 are utilized. The material properties discussed in

6

this example are assumed to vary continuously along the radial direction according to

7

trigonometric functions formulated in Eqs. (40) and (41). The normalized deflections

8

are denoted as u z  u z

9

non-dimensional deflections at the interesting point A under different gradient

10

coefficients, boundary conditions and thickness-to-length ratios are listed in Tables

11

19-20. Moreover, Figs. 11-14 display variations of the dimensionless transverse

12

displacements through the radius direction from (Ri, 0, 0) to (Ro, 0, 0) with α=0.3.

13

Both for the simply supported and clamped plates, it is observed that the thinner the

14

plate is, the smaller the transverse displacement is. Moreover, with increasing the

15

gradient index, the normalized deflections increase. When the boundary constraint

16

alters from simple support to fixed support which means that the flexural rigidity of

17

the perforated plate increases, the magnitudes of the elastic displacements in z

18

direction reduce. In Figs. 11-12, the deflection curves almost coincide with each other

19

irrespective of gradation functions. In other words, changes of aspect ratios may

20

impose little influence on variations of deflections through the radial direction for the

21

simply supported plate. Figs. 13-14 exhibit that increasing the thickness-to-length

22

ratios for both models decreases the amplitudes of dimensionless deflections. What’s

23

more, it is apparent that the transverse displacements monotonically decrease with the

24

locations moving from the inner boundary to the outer boundary as expected. A

25

conclusion can be drawn that the boundary constraints and gradient indexes make

26

remarkable influences on the deformable behaviors of the perforated in-plane

27

functionally graded plate subjected to uniform external forces.

28

10 3 D ql 4

with the bending stiffness

Model 1: 18

D

E0t 3 . The 12 1   02 

1

2

P  r   P0 1   cos  r Ro   2  

(40)

P  r   P0 1   cos  r 2 Ro  

(41)

Model 2:

5. Conclusions

3

For the first time, the SBFEM in conjunction with the PIA are utilized to

4

investigate the distributions of displacement and stress components in the in-plane

5

functionally graded plates. The displacements along the thickness direction are

6

expressed as the analytical matrix exponential. The PIA is employed to compute the

7

displacements and stresses, which makes sure that predicted results can reach up to

8

the limited precision of the computer used. Aided by the comparisons with exact

9

three-dimensional elasticity and numerical solutions available in the literature, the

10

high accuracy, reliability, effectiveness and applicability of the introduced

11

semi-analytical approach to explore the mechanical behaviors of in-plane functionally

12

graded plates have been established. Other numerical examples are provided to

13

demonstrate the effect of boundary constraints, aspect ratios and gradient indexes. The

14

corresponding conclusions are in the following:

15

1. In the circular plate, with increasing the thickness-to-length ratios, the amplitudes

16

of dimensionless displacements and stresses decrease. The deflections almost

17

remain constants along the thickness but the maxima of ux and σxx occur at the

18

bottom and top planes. But for τxy, the distribution curve for S=10 is quite

19

different from those for S=50 and 100.

20

2. Regarding the perforated plate, it is observed that the thinner the plat is, the smaller

21

the transverse displacement is. Moreover, with increasing the gradient index, the

22

normalized deflections increase. For the simply supported plate, the deflection

23

curves along the radial direction almost coincide with each other irrespective of

24

gradation functions. But in the clamped plate, increasing the thickness-to-length

25

ratios decreases the amplitudes of dimensionless deflections. Additionally, the

26

transverse displacements monotonically decrease with the locations moving from

27

the inner boundary to the outer boundary. The forthcoming papers will reveal 19

1 2 3

more meaningful results.

CRediT authorship contribution statement Pengchong

Zhang:

Conceptualization,

Methodology,

Computation,

4

Writing-original draft, Data analysis, Writing-review & editing. Chengzhi Qi:

5

Conceptualization, Supervision. Hongyuan Fang: Conceptualization, Supervision.

6

Wei He: Supervision.

7

Declaration of Competing Interest

8

The authors declare that they have no known competing financial interests or

9

personal relationships that could have appeared to influence the work reported in this

10

paper.

11

Acknowledgement

12

This research is supported by Grants 2018M641168 from China Postdoctoral

13

Science Foundation, Grants 51908022, 51478027, 2015CB57800 and 51774018 from

14

the National Natural Science Foundation of China, Grant IRT_17R06 from program

15

for Changjiang Scholars and Innovative Research Team, for which the authors are

16

gratefully acknowledged.

17

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17 18 19 20 21 22 23 24 25 26 27 28 29 30 27

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

The captions of all figures are listed as follows:

16

Fig. 1 A model of the in-plane functionally graded plate.

17

Fig. 2 A square plate meshed with four spectral elements.

18

Fig. 3 Distributions of the in-plane displacements for the 0° plate: (a) ux; (b) uy.

19

Fig. 4 Variations of the in-plane displacements of the 90° plate: (a) ux; (b) uy.

20

Fig. 5 The circular in-plane functionally graded plate: (a) the plate model; (b) discretized

21

with twelve fourth-order spectral elements.

22

Fig. 6 Cross-thickness distributions of the normalized deflections uz.

23

Fig. 7 Cross-thickness distributions of the elastic displacement ux.

24

Fig. 8 Cross-thickness distributions of the normal stress σxx.

25

Fig. 9 Cross-thickness distributions of the shear stress τxy.

26

Fig. 10 The plane view of the perforated plate: (a) the geometry of the plate; (b) meshed with

27

four spectral elements.

28

Fig. 11 Variations of the deflections for the simply supported plate with Model 1.

29

Fig. 12 Distributions of the deflections for the simply supported plate with Model 2.

30

Fig. 13 Variations of the deflections for the clamped plate with Model 1. 28

1

Fig. 14 Distributions of the deflections for the clamped plate with Model 2.

2 3 4 5 6 7 8 9 10 11 12 13 14 15

The captions of all tables are listed as follows:

16

Table 1 The convergence analysis of deflections for the CSFS plate with ky=1 and Model 1.

17

Table 2 The convergence study of deflections for the CSFS plate with ky=1 and Model 2.

18

Table 3 The normalized deflections of the CCCC plate with Model 1.

19

Table 4 The normalized deflections of the CCCC plate with Model 2.

20

Table 5 The non-dimensional deflections of the SSSS plate with Model 1.

21

Table 6 The non-dimensional deflections of the SSSS plate with Model 2.

22

Table 7 The dimensionless deflections of the CSFS plate with Model 1.

23

Table 8 The dimensionless deflections of the CSFS plate with Model 2.

24

Table 9 The convergence study of displacements and normal stresses for the 0° plate.

25

Table 10 The convergence analysis of displacements and normal stresses for the 90° plate.

26

Table 11 Variations of elastic displacements and normal stresses for the 0° plate.

27

Table 12 Variations of shear stresses for the 0° plate.

28

Table 13 Distributions of elastic displacements and normal stresses for the 90° plate.

29

Table 14 Distributions of shear stresses for the 90° plate.

30

Table 15 Displacements and normal stresses along the thickness for the 0° plate with l/t=5. 29

1

Table 16 Shear stresses along the thickness for the 0° plate with l/t=5.

2

Table 17 Through-thickness displacements and normal stresses for the 90° plate with l/t=5.

3

Table 18 Through-thickness shear stresses for the 90° plate with l/t=5.

4

Table 19 The dimensionless transverse displacements of the simply supported plate.

5

Table 20 The dimensionless transverse displacements of the clamped plate.

6

CRediT authorship contribution statement

7

Pengchong

Zhang:

Conceptualization,

Methodology,

Computation,

8

Writing-original draft, Data analysis, Writing-review & editing. Chengzhi Qi:

9

Conceptualization, Supervision. Hongyuan Fang: Conceptualization, Supervision.

10

Wei He: Supervision.

11 12

Declaration of Competing Interest

13

The authors declare that they have no known competing financial interests or

14

personal relationships that could have appeared to influence the work reported in this

15

paper.

16 17

Table 1 The convergence analysis of deflections for the CSFS plate with ky=1 and Model 1. S=l/t

5

50

100

Elem. order 2 3 4 Ref. [52] 4 6 8 10 Ref. [52] 4 6 8 10 Ref. [52]

kx=0.0

kx=0.5

kx=1.0

kx=2.0

0.2311 0.2822 0.2881 0.2857 2.1743 2.1871 2.1959 2.2035 2.2523 4.3349 4.3538 4.3637 4.3706 4.4938

0.2568 0.3107 0.3171 0.3145 2.3775 2.3954 2.4074 2.4172 2.4690 4.7389 4.7665 4.7813 4.7912 4.9258

0.2660 0.3246 0.3314 0.3283 2.4838 2.5045 2.5173 2.5273 2.5691 4.9506 4.9840 5.0006 5.0111 5.1255

0.2776 0.3406 0.3475 0.3424 2.6019 2.6234 2.6366 2.6469 2.6736 5.1861 5.2208 5.2379 5.2486 5.3338

18 19 20

Table 2 The convergence study of deflections for the CSFS plate with ky=1 and Model 2. 30

S=l/t

5

50

100

Elem. order 2 3 4 Ref. [52] 4 6 8 10 Ref. [52] 4 6 8 10 Ref. [52]

kx=0.0

kx=0.5

kx=1.0

kx=2.0

0.2181 0.2644 0.2708 0.2666 2.0457 2.0584 2.0658 2.0724 2.1058 4.0721 4.0984 4.1064 4.1120 4.2019

0.2456 0.2948 0.3028 0.2999 2.2735 2.3003 2.3111 2.3195 2.3702 4.4932 4.5772 4.5895 4.5967 4.7291

0.2551 0.3102 0.3186 0.3154 2.3962 2.4301 2.4363 2.4458 2.4838 4.7293 4.8233 4.8396 4.8493 4.9553

0.2683 0.3286 0.3366 0.3312 2.5286 2.5597 2.5657 2.5756 2.5977 5.0064 5.0804 5.0970 5.1073 5.1822

1 2 3

Table 3 The normalized deflections of the CCCC plate with Model 1.

S=l/t

5

50

100

ky 0.0 0.5 1.0 2.0 0.0 0.5 1.0 2.0 0.0 0.5 1.0 2.0

kx=0.0 Ref. [52] SBFEM 0.0823 0.0823 0.0983 0.0987 0.1079 0.1068 0.1191 0.1172 0.5199 0.5202 0.6124 0.6074 0.6655 0.6586 0.7278 0.7238 1.0346 1.0342 1.2185 1.1910 1.3239 1.2918 1.4476 1.4197

kx=0.5 Ref. [52] SBFEM 0.0983 0.0987 0.1108 0.1102 0.1179 0.1165 0.1259 0.1244 0.6124 0.6074 0.6837 0.6741 0.7228 0.7148 0.7673 0.7642 1.2185 1.1910 1.3601 1.3209 1.4378 1.4012 1.5261 1.4982

kx=1.0 Ref. [52] SBFEM 0.1079 0.1068 0.1179 0.1165 0.1235 0.1219 0.1298 0.1283 0.6655 0.6586 0.7228 0.7148 0.7540 0.7484 0.7889 0.7881 1.3239 1.2918 1.4378 1.4012 1.4997 1.4671 1.5691 1.5451

kx=2.0 Ref. [52] SBFEM 0.1191 0.1172 0.1259 0.1244 0.1298 0.1283 0.1341 0.1330 0.7278 0.7238 0.7673 0.7642 0.7889 0.7881 0.8128 0.8159 1.4476 1.4197 1.5261 1.4982 1.5691 1.5451 1.6165 1.5995

4 5 6

Table 4 The normalized deflections of the CCCC plate with Model 2.

S=l/t

5

50

100

ky 0.0 0.5 1.0 2.0 0.0 0.5 1.0 2.0 0.0 0.5 1.0 2.0

kx=0.0 Ref. [52] SBFEM 0.0823 0.0823 0.0957 0.0951 0.1027 0.1044 0.1108 0.1132 0.5199 0.5202 0.5996 0.5969 0.6370 0.6350 0.6802 0.6797 1.0346 1.0342 1.1930 1.1695 1.2672 1.2444 1.3530 1.3319

kx=0.5 Ref. [52] SBFEM 0.0957 0.0951 0.1078 0.1082 0.1135 0.1139 0.1198 0.1212 0.5996 0.5969 0.6700 0.6614 0.6991 0.6937 0.7307 0.7285 1.1930 1.1695 1.3329 1.2937 1.3907 1.3574 1.4533 1.4259 31

kx=1.0 Ref. [52] SBFEM 0.1027 0.1044 0.1135 0.1139 0.1185 0.1190 0.1237 0.1254 0.6370 0.6350 0.6991 0.6937 0.7236 0.7214 0.7497 0.7502 1.2672 1.2444 1.3907 1.3574 1.4392 1.4121 1.4909 1.4689

kx=2.0 Ref. [52] SBFEM 0.1108 0.1132 0.1198 0.1212 0.1237 0.1254 0.1278 0.1305 0.6802 0.6797 0.7307 0.7285 0.7497 0.7502 0.7696 0.7687 1.3530 1.3319 1.4533 1.4259 1.4909 1.4689 1.5304 1.5125

1 2 3

Table 5 The non-dimensional deflections of the SSSS plate with Model 1.

S=l/t

5

50

100

ky 0.0 0.5 1.0 2.0 0.0 0.5 1.0 2.0 0.0 0.5 1.0 2.0

kx=0.0 Ref. [52] SBFEM 0.1752 0.1763 0.2075 0.2069 0.2264 0.2275 0.2485 0.2476 1.4542 1.4553 1.7083 1.6718 1.8568 1.8148 2.0284 1.9932 2.9039 2.9045 3.4111 3.3311 3.7074 3.6159 4.0498 3.9711

kx=0.5 Ref. [52] SBFEM 0.2075 0.2069 0.2324 0.2322 0.2467 0.2469 0.2628 0.2620 1.7083 1.6718 1.9046 1.8521 2.0166 1.9686 2.1429 2.1086 3.4111 3.3311 3.8028 3.6908 4.0264 3.9227 4.2784 4.2013

kx=1.0 Ref. [52] SBFEM 0.2264 0.2275 0.2467 0.2469 0.2579 0.2590 0.2703 0.2702 1.8568 1.8148 2.0166 1.9686 2.1043 2.0638 2.2011 2.1745 3.7074 3.6159 4.0264 3.9227 4.2013 4.1123 4.3945 4.3325

kx=2.0 Ref. [52] SBFEM 0.2485 0.2476 0.2628 0.2620 0.2703 0.2702 0.2783 0.2795 2.0284 1.9932 2.1429 2.1086 2.2011 2.1745 2.2629 2.2470 4.0498 3.9711 4.2784 4.2013 4.3945 4.3325 4.5180 4.4770

4 5 6

Table 6 The non-dimensional deflections of the SSSS plate with Model 2.

S=l/t

5

50

100

ky 0.0 0.5 1.0 2.0 0.0 0.5 1.0 2.0 0.0 0.5 1.0 2.0

kx=0.0 Ref. [52] SBFEM 0.1752 0.1763 0.2043 0.2048 0.2199 0.2197 0.2380 0.2384 1.4542 1.4553 1.6841 1.6501 1.8074 1.7717 1.9494 1.9203 2.9039 2.9045 3.3627 3.2888 3.6089 3.5313 3.8922 3.8275

kx=0.5 Ref. [52] SBFEM 0.2043 0.2048 0.2242 0.2232 0.2356 0.2369 0.2492 0.2491 1.6841 1.6501 1.8335 1.7884 1.9202 1.8784 2.0253 1.9925 3.3627 3.2888 3.6607 3.5648 3.8338 3.7440 4.0434 3.9711

kx=1.0 Ref. [52] SBFEM 0.2199 0.2197 0.2356 0.2369 0.2448 0.2439 0.2559 0.2562 1.8074 1.7717 1.9202 1.8784 1.9888 1.9499 2.0743 2.0434 3.6089 3.5313 3.8338 3.7440 3.9707 3.8863 4.1412 4.0723

kx=2.0 Ref. [52] SBFEM 0.2380 0.2384 0.2492 0.2491 0.2559 0.2562 0.2642 0.2654 1.9494 1.9203 2.0253 1.9925 2.0743 2.0434 2.1377 2.1128 3.8922 3.8275 4.0434 3.9711 4.1412 4.0723 4.2677 4.2104

7 8 9

Table 7 The dimensionless deflections of the CSFS plate with Model 1.

S=l/t

5

50 100

ky 0.0 0.5 1.0 2.0 0.0 0.5 1.0 2.0 0.0 0.5

kx=0.0 Ref. [52] SBFEM 0.2188 0.2238 0.2595 0.2578 0.2857 0.2881 0.3170 0.3144 1.7490 1.7500 2.0552 2.0158 2.2523 2.2035 2.4873 2.4490 3.4903 3.4904 4.1009 3.9991

kx=0.5 Ref. [52] SBFEM 0.2651 0.2640 0.2954 0.2975 0.3145 0.3171 0.3365 0.3347 2.0967 2.0618 2.3241 2.2691 2.4690 2.4172 2.6365 2.6031 4.1837 4.0858 4.6371 4.4973 32

kx=1.0 Ref. [52] SBFEM 0.2887 0.2912 0.3131 0.3150 0.3283 0.3314 0.3455 0.3442 2.2673 2.2301 2.4516 2.4040 2.5691 2.5273 2.7035 2.6796 4.5237 4.4219 4.8912 4.7667

kx=2.0 Ref. [52] SBFEM 0.3141 0.3113 0.3316 0.3289 0.3424 0.3475 0.3544 0.3547 2.4538 2.4237 2.5880 2.5547 2.6736 2.6469 2.7705 2.7589 4.8957 4.8066 5.1632 5.0660

1.0 2.0

4.4938 4.9623

4.3706 4.8564

4.9258 5.2599

4.7912 5.1598

5.1255 5.3935

5.0111 5.3126

5.3338 5.5270

5.2486 5.4704

1 2 3

Table 8 The dimensionless deflections of the CSFS plate with Model 2.

S=l/t

5

50

100

ky 0.0 0.5 1.0 2.0 0.0 0.5 1.0 2.0 0.0 0.5 1.0 2.0

kx=0.0 Ref. [52] SBFEM 0.2188 0.2238 0.2485 0.2476 0.2666 0.2708 0.2894 0.2877 1.7490 1.7500 1.9704 1.9436 2.1058 2.0724 2.2774 2.2479 3.4903 3.4904 3.9319 3.8568 4.2019 4.1120 4.5440 4.4594

kx=0.5 Ref. [52] SBFEM 0.2631 0.2609 0.2864 0.2887 0.2999 0.3028 0.3162 0.3130 2.0934 2.0558 2.2699 2.2150 2.3702 3.3195 2.4910 2.4532 4.1771 4.0719 4.5290 4.3889 4.7291 4.5967 4.9699 4.8621

kx=1.0 Ref. [52] SBFEM 0.2853 0.2889 0.3045 0.3069 0.3154 0.3186 0.3284 0.3252 2.2554 2.2221 2.4025 2.3591 2.4838 2.4458 2.5799 2.5537 4.4998 4.4041 4.7931 4.6769 4.9553 4.8493 5.1470 5.0632

kx=2.0 Ref. [52] SBFEM 0.3089 0.3064 0.3232 0.3199 0.3312 0.3366 0.3406 0.3397 2.4270 2.4055 2.5378 2.5109 2.5977 2.5756 2.6675 2.6542 4.8416 4.7687 5.0627 4.9788 5.1822 5.1073 5.3213 5.2632

4 5 6

Table 9 The convergence study of displacements and normal stresses for the 0° plate. S=l/t

20

50

100

Elem. order 4 6 8 10 Ref. [56] 4 6 8 10 Ref. [56] 4 6 8 10 Ref. [56]

uz(l/2,l/2,0)

-σxx (l/2,l/2,-t/2)

-σyy(l/2,l/2,-t/2)

0.2090 0.2087 0.2089 0.2091 0.209 0.1712 0.1739 0.1741 0.1740 0.174 0.1691 0.1689 0.1691 0.1690 0.169

59.4952 57.2973 58.1252 57.9148 58.0 58.5122 56.3662 57.2871 57.0069 57.0 58.3776 56.2344 57.1660 56.8588 56.9

2.0835 2.0897 2.0970 2.1002 2.10 1.7517 1.7565 1.7637 1.7609 1.76 1.7036 1.7155 1.7083 1.7128 1.71

7 8 9

Table 10 The convergence analysis of displacements and normal stresses for the 90° plate. S=l/t

20

Elem. order 4 6 8 10

uz(l/2,l/2,0)

-σxx (l/2,l/2,-t/2)

-σyy(l/2,l/2,-t/2)

0.1879 0.1864 0.1864 0.1865

1.7734 1.6755 1.6820 1.6808

62.0162 62.4440 62.5526 62.5887

33

50

100

Ref. [56] 4 6 8 10 Ref. [56] 4 6 8 10 Ref. [56]

0.186 0.1642 0.1624 0.1623 0.1623 0.162 0.1608 0.1589 0.1589 0.1589 0.159

1.72 1.6596 1.7099 1.5325 1.5318 1.54 1.6538 1.5748 1.5112 1.5111 1.51

62.6 60.9108 61.1816 61.1944 61.2020 61.2 60.7510 60.9806 60.9969 60.9985 61.0

1 2 3

Table 11 Variations of elastic displacements and normal stresses for the 0° plate. S=l/t 5 10 20 50 100 CPT

uz(l/2,l/2,0) Ref. [56] SBFEM 0.752 0.7670 0.329 0.3298 0.209 0.2090 0.174 0.1740 0.169 0.1690 0.167 0.1690

-σxx(l/2,l/2,-t/2) Ref. [56] 74.6 61.6 58.0 57.0 56.9 56.8

SBFEM 75.3566 61.8140 57.9148 57.0069 56.8588 56.8588

-σyy(l/2,l/2,-t/2) Ref. [56] 6.97 3.23 2.10 1.76 1.71 1.70

SBFEM 6.9831 3.2297 2.1002 1.7609 1.7083 1.7083

4 5 6

Table 12 Variations of shear stresses for the 0° plate. S=l/t 5 10 20 50 100 CPT

-τxy(0,0,t/2) Ref. [56] 2.79 1.43 0.972 0.823 0.798 0.790

SBFEM 2.8544 1.3988 0.9662 0.8231 0.7981 0.7981

τxz(0,l/2,0) Ref. [56] 34.4 40.4 43.3 44.7 45.1 45.0

SBFEM 31.4795 40.4384 43.4855 44.4827 44.7529 44.7529

τyz(l/2,0,0) Ref. [56] 7.13 3.92 2.89 2.58 2.53 2.52

SBFEM 7.2327 4.1765 2.9086 2.6078 2.5384 2.5384

7 8 9

Table 13 Distributions of elastic displacements and normal stresses for the 90° plate. S=l/t 5 10 20 50 100 CPT

uz(l/2,l/2,0) Ref. [56] SBFEM 0.576 0.5812 0.270 0.2708 0.186 0.1864 0.162 0.1623 0.159 0.1589 0.158 0.1588

-σxx(ξ,l/2,-t/2) Ref. [56] SBFEM 4.81 4.7124 2.35 2.3507 1.72 1.6820 1.54 1.5325 1.51 1.5111 1.50 1.5111

-σyy(l/2,l/2,-t/2) Ref. [56] SBFEM 82.3 82.3143 67.0 67.0670 62.6 62.5887 61.2 61.2020 61.0 60.9985 60.9 60.9256

10 11 12

Table 14 Distributions of shear stresses for the 90° plate. 34

S=l/t 5 10 20 50 100 CPT

-τxy(0,0,t/2) Ref. [56] 2.05 1.26 0.985 0.883 0.864 0.859

SBFEM 2.0336 1.2346 0.9838 0.8836 0.8641 0.8645

τxz(0,l/2,0) Ref. [56] 6.88 4.80 4.17 4.00 3.98 4.00

SBFEM 6.9503 5.1587 4.1811 3.9814 4.0049 4.0049

τyz(l/2,0,0) Ref. [56] 41.7 46.7 48.2 48.5 48.6 48.6

SBFEM 41.7744 46.3894 48.1644 48.5903 48.5996 48.5996

1 2 3

Table 15 Displacements and normal stresses along the thickness for the 0° plate with l/t=5. ux(0,l/2,z) Ref. [56] SBFEM 0.422 0.42516 0.197 0.19798 0.0926 0.09272 0.0404 0.04042 0.0128 0.01279 -0.00477 0.00476 -0.0220 -0.02196 -0.0483 -0.04827 -0.0975 -0.09758 -0.196 -0.19666 -0.407 -0.41009

z/t -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

uy(l/2,0, z) Ref. [56] SBFEM 0.897 0.8996 0.665 0.6673 0.451 0.4524 0.250 0.2505 0.0574 0.0575 -0.130 -0.1304 -0.316 -0.3170 -0.504 -0.5060 -0.699 -0.7013 -0.904 -0.9067 -1.12 -1.1265

σxx(l/2,l/2,z) Ref. [56] SBFEM -74.6 -74.5151 -41.4 -41.5091 -21.5 -21.5744 -10.4 -10.4092 -4.03 -4.0330 0.270 0.2711 4.48 4.4907 10.6 10.5781 21.1 21.1377 39.8 39.8825 71.2 71.1431

σyy(l/2,l/2,z) Ref. [56] SBFEM -6.97 -6.9724 -5.52 -5.5183 -4.13 -4.1321 -2.80 -2.8007 -1.51 -1.5103 -0.242 -0.2409 1.02 1.0187 2.28 2.2857 3.57 3.5755 4.90 4.9028 6.28 6.2804

σzz(l/2,l/2,z) Ref. [56] SBFEM -1 -1.0000 -0.970 -0.9700 -0.888 -0.8885 -0.773 -0.7730 -0.639 -0.6389 -0.497 -0.4969 -0.355 -0.3555 -0.223 -0.2227 -0.109 -0.1092 -0.0293 -0.0294 0 -0.0000

4 5 6

Table 16 Shear stresses along the thickness for the 0° plate with l/t=5. z/t -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

τxy(0,0,z) Ref. [56] SBFEM 2.37 2.4133 1.62 1.6521 1.05 1.0765 0.574 0.5893 0.145 0.1482 -0.262 -0.2715 -0.666 -0.6878 -1.09 -1.1188 -1.55 -1.5879 -2.08 -2.1353 -2.79 -2.8517

τxz(0,l/2,z) Ref. [56] SBFEM 0 0.0000 20.7 20.7245 28.8 28.6603 32.5 32.2301 34.1 33.7375 34.4 34.0323 33.7 33.3577 31.8 31.5118 27.8 27.7109 19.8 19.8085 0 0.0000

τyz(l/2,0,z) Ref. [56] SBFEM 0 0.0000 2.71 2.7418 4.71 4.7609 6.08 6.1446 6.87 6.9477 7.13 7.2008 6.84 6.9158 6.02 6.0883 4.64 4.6959 2.66 2.6923 0 0.0000

7 8 9 z/t -0.5 -0.4 -0.3 -0.2

Table 17 Through-thickness displacements and normal stresses for the 90° plate with l/t=5. ux(0,l/2,z) Ref. [56] SBFEM 0.754 0.7560 0.539 0.5404 0.349 0.3497 0.176 0.1766

uy(l/2,0, z) Ref. [56] SBFEM 0.327 0.3275 0.182 0.1822 0.0982 0.0982 0.0490 0.0490

σxx(l/2,l/2,z) Ref. [56] SBFEM -4.81 -4.8099 -3.86 -3.8566 -2.93 -2.9266 -2.02 -2.0190 35

σyy(l/2,l/2,z) Ref. [56] SBFEM -82.3 -82.3326 -46.6 -46.6019 -25.7 -25.7454 -13.3 -13.3298

σzz(l/2,l/2,z) Ref. [56] SBFEM -1 -1.0000 -0.964 -0.9638 -0.877 -0.8770 -0.762 -0.7620

-0.1 0.0 0.1 0.2 0.3 0.4 0.5

0.0153 -0.139 -0.292 -0.448 -0.612 -0.790 -0.987

0.0153 -0.1397 -0.2929 -0.4494 -0.6139 -0.7917 -0.9887

0.0184 -0.00385 -0.0256 -0.0545 -0.100 -0.178 -0.312

0.0184 -0.0039 -0.0257 -0.0546 -0.1005 -0.1783 -0.3129

-1.13 -0.254 0.617 1.49 2.36 3.24 4.14

-1.1293 -0.2519 0.6190 1.4894 2.3645 3.2485 4.1425

-5.48 0.349 6.05 13.5 25.1 44.4 77.5

-5.4742 0.3519 6.0764 13.4925 25.1258 44.4218 77.5236

-0.631 -0.495 -0.359 -0.231 -0.118 -0.0345 0

1 2 3

Table 18 Through-thickness shear stresses for the 90° plate with l/t=5. z/t -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

τxy(0,0,z) Ref. [56] SBFEM 1.72 1.7244 1.17 1.1743 0.730 0.7361 0.370 0.3746 0.0605 0.0613 -0.224 -0.2273 -0.505 -0.5123 -0.803 -0.8141 -1.14 -1.1550 -1.55 -1.5606 -2.05 -2.0620

τxz(0,l/2,z) Ref. [56] SBFEM 0 0.0000 2.81 2.8418 4.73 4.7708 5.99 6.0147 6.68 6.6944 6.88 6.8725 6.59 6.5737 5.82 5.7922 4.53 4.4925 2.64 2.6028 0 0.0000

τyz(l/2,0,z) Ref. [56] SBFEM 0 0.0000 20.4 20.7244 31.9 32.2484 38.1 38.4647 41.0 41.4191 41.7 42.1363 40.6 40.9517 37.2 37.6154 30.8 31.2140 19.5 19.8742 0 0.0000

4 5 6

Table 19 The dimensionless transverse displacements of the simply supported plate.

Model 1

Model 2

S=l/t

α=0.1

α=0.2

α=0.3

α=0.4

α=0.5

10

1.5937

1.7543

1.9457

2.1776

2.4647

20

1.5902

1.7508

1.9421

2.1739

2.4609

50

1.5893

1.7498

1.9411

2.1729

2.4599

100

1.5891

1.7497

1.9409

2.1727

2.4597

10

1.5429

1.6382

1.7443

1.8630

1.9970

20

1.5396

1.6349

1.7409

1.8596

1.9936

50

1.5386

1.6339

1.7400

1.8587

1.9926

100

1.5385

1.6338

1.7398

1.8586

1.9925

7 8 9

Table 20 The dimensionless transverse displacements of the clamped plate.

Model 1

Model 2

S=l/t

α=0.1

α=0.2

α=0.3

α=0.4

α=0.5

10

0.1413

0.1451

0.1493

0.1540

0.1592

20

0.1174

0.1208

0.1246

0.1287

0.1333

50

0.1084

0.1117

0.1153

0.1193

0.1238

100

0.1070

0.1103

0.1139

0.1179

0.1223

10

0.1399

0.1422

0.1445

0.1470

0.1497

20

0.1161

0.1181

0.1202

0.1224

0.1247

36

-0.6311 -0.4947 -0.3588 -0.2307 -0.1180 -0.0346 0.0000

50

0.1071

0.1090

0.1110

0.1131

0.1154

100

0.1057

0.1076

0.1096

0.1117

0.1139

1 2

3

4

37

1

2

38

1

2

39

1

2

40

1

2

41

1

2

42

1

2

43

1

2

44

1

2

45