Polyharmonic (thin-plate) splines in the analysis of composite plates

ARTICLE IN PRESS

International Journal of Mechanical Sciences 46 (2004) 1549–1569 www.elsevier.com/locate/ijmecsci

Polyharmonic (thin-plate) splines in the analysis of composite plates A.J.M. Ferreira Departamento de Engenharia Mecaˆnica e Gesta˜o Industrial, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias 4200-465 Porto, Portugal Received 1 October 2003; received in revised form 3 September 2004; accepted 5 September 2004 Available online 27 October 2004

Abstract In the present study a layerwise theory for composite and sandwich laminated plates is discretized by polyharmonic (thin-plates) splines. A composite and a sandwich plate examples are presented and discussed. The combination of adequate shear deformation theory and thin-plate splines allows a very accurate prediction of displacements and stresses. r 2004 Elsevier Ltd. All rights reserved. Keywords: Polyharmonic splines; Radial basis functions; Composite plates; Layerwise theory

1. Introduction Composite plates are one of the most significant applications of composite materials in industry. Layers are stacked together to form thin or thick laminates. Several laminate theories have been applied to composite laminates. The classical laminated plate theory (CLPT) is an extension of the Love–Kirchhoff assumptions for isotropic plates and can be applied if the laminate is thin because transverse shear stresses [1–6] are not considered. The first-order shear deformation theory (FSDT) [7–11] which accounts for transverse shear effects, needs some shearcorrection factors [7,12,13]. Tel.: +351 22 957 8713; fax: +351 22 953 7352.

E-mail address: [email protected] (A.J.M. Ferreira). 0020-7403/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2004.09.002

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The classical laminate plate theory and the first-order shear deformation theory describe with reasonable accuracy the kinematics of most laminates. Higher-order theories [14,13] can represent the kinematics better, may disregard shear-correction factors and yield more accurate transverse shear stresses. All such theories consider the same degrees of freedom for all laminate layers. In some cases, particularly in sandwich applications, the difference between material properties makes it difficult for such theories to fully accommodate the bending behavior. In fact most of them do not correctly represent the transverse shear stresses. Another set of theories that were introduced back in the 1980s are the layerwise theories, that consider degrees of freedom for each layer. The layerwise theory of Reddy [13] is perhaps the most popular layerwise theory for composite and sandwich plate analysis. In this work we adopt a somewhat different kind of layerwise theory, based on an expansion of Mindlin’s theory in each layer. The displacement continuity at layer’s interface is guaranteed. Also the theory produces directly very accurate transverse shear stress, although constant, in each layer. Most of the spatial discretization techniques so far have been based on finite differences and finite elements. In the present study it is implemented a layerwise theory in a mesh-free discretization based on polyharmonic (thin-plate) splines radial basis functions. This truly meshless technique insensitive to spatial dimension, considers only a cloud of nodes (centers) for the spatial discretization of both the problem domain and the boundary. Other meshless methods have also been proposed. They may be classified as smooth particle hidrodynamics [15], diffuse element method [16], element-free galerkin [17–19], reproducing kernel particle method [20–25] and hp clouds [26]. The radial basis function method was first used by Hardy [27,28] for the interpolation of geographical scattered data and later used by Kansa [29,30] for the solution of partial differential equations (PDEs). Many other radial basis functions can be used as reviewed in the recent book of Liu [31], as in Powell [32], Coleman [33], Sharan et al. [34], Wendland [35], among others. The method has also been applied to other engineering problems such as in [36–38]. The use of radial basis functions (RBFs) for 2-D solids has been proposed by Liu et al. [39–41] and by Ferreira [12,42] for composite laminated plates and beams using the first-order shear deformation theory. This paper focuses, for the first time, on the analysis of composite laminated plates by the combined use of thin-plate splines and refined shear deformation theories.

2. Polyharmonic (thin-plate) splines formulation The multiquadric radial basis function method has been used by the author in laminated beams and plates. However, such formulations rely on a shape parameter for which a theoretical background still is not completely established. Thin-plate splines do not need the use of such parameter which makes them very attractive for structural applications. RBFs’ splines depend only on the distance to a center point xj and is of the form gðkx  xj kÞ [29,30,43–46]. Consider a set of nodes x1 ; x2 ; . . . ; xN 2 O  Rn : The radial basis functions centered at xj are defined as gj ðxÞ  gðkx  xj kÞ 2 Rn ;

j ¼ 1; . . . ; N;

(1)

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where kx  xj k is the Euclidian norm. The polyharmonic thin-plate splines functions are of the form Thin plate splines : gj ðxÞ ¼ kx  xj k2m log kx  xj k;

m ¼ 1; 2; . . . :

(2)

In this paper we used gj ðxÞ ¼ kx  xj k6 log kx  xj k

(3)

due to the need of second-order differentiation. One of the main advantages of radial basis functions is the insensitivity to spatial dimension, making the implementation of this method much easier than, e.g., finite elements [29,30]. The method does not require a grid or mesh, being the only geometric properties the pairwise distances between points. In this paper it is proposed to use Kansa’s unsymmetric collocation method [29,30]. Consider a boundary-valued problem with a domain O  Rn and a linear elliptic partial differential equation of the form LuðxÞ ¼ sðxÞ  Rn ;

(4)

BuðxÞjqO ¼ f ðxÞ 2 Rn ;

(5)

where qO represents the boundary of the problem. We use points along the boundary ðxj ; j ¼ 1; . . . ; N B Þ and in the interior ðxj ; j ¼ N B þ 1; . . . ; NÞ: Let the RBF interpolant to the solution uðxÞ be sðx; cÞ ¼

N X

aj gðkx  xj kÞ:

(6)

j¼1

Collocation with the boundary data at the boundary points and with PDE at the interior points leads to equations sB ðx; cÞ 

N X

aj Bgðkx  xj kÞ ¼ lðxi Þ;

i ¼ 1; . . . ; N B ;

(7)

aj Lgðkx  xj kÞ ¼ Fðxi Þ;

i ¼ N B þ 1; . . . ; N;

(8)

j¼1

sL ðx; cÞ 

N X j¼1

where lðxi Þ; Fðxi Þ are the prescribed values at the boundary nodes and the function values at the interior nodes, respectively. This corresponds to a system of equations with an unsymmetric coefficient matrix, structured in matrix form as     Bg l ½a ¼ : (9) Lg U The use of globally supported RBFs for large problems can bring problems due to the full populated matrices. To solve this drawback a localization scheme is advisable. Domain

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decomposition methods [47,43], localization of the basis functions [48,43] claim to be able to deal with tens of thousands of nodes.

3. A layerwise theory The layerwise proposed in this paper is based on the assumption of a first-order shear deformation theory in each layer and the imposition of displacement continuity at layer’s interfaces. In each layer the same assumptions than the first-order plate theories are considered. Due to the size and complexity of the formulation we restrict the analysis to a three-layer laminate, as shown schematically in Fig. 1. The displacement field for the middle layer (sometimes known as the core in sandwich laminates) is given as uð2Þ ðx; y; zÞ ¼ u0 ðx; yÞ þ zð2Þ fð2Þ x ;

(10)

vð2Þ ðx; y; zÞ ¼ v0 ðx; yÞ þ zð2Þ fð2Þ y ;

(11)

wð2Þ ðx; y; zÞ ¼ w0 ðx; yÞ;

(12)

where u and v are the in-plane displacements at any point ðx; y; zÞ; u0 and v0 denote the in-plane ð2Þ displacement of the point ðx; y; 0Þ on the mid-plane, w is the deflection, fð2Þ x and fy are the rotations of the normals to the mid-plane about the y and x axes, respectively, for layer 2 (middle layer).

z

x(3)

h3

z(3)

3

x(2)

h2

z(2)

2 (1)

x

h1

z(1)

1 layers

Fig. 1. 1-D representation of the layerwise kinematics.

x

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The correspondent displacement field for the upper layer (3) and lower layer (1) are given, respectively, as uð3Þ ðx; y; zÞ ¼ u0 ðx; yÞ þ

h2 ð2Þ h3 ð3Þ fx þ fx þ zð3Þ fð3Þ x ; 2 2

(13)

vð3Þ ðx; y; zÞ ¼ v0 ðx; yÞ þ

h2 ð2Þ h3 ð3Þ f þ fy þ zð3Þ fð3Þ y ; 2 y 2

(14)

wð3Þ ðx; y; zÞ ¼ w0 ðx; yÞ;

(15)

uð1Þ ðx; y; zÞ ¼ u0 ðx; yÞ 

h2 ð2Þ h1 ð1Þ f  fx þ zð1Þ fð1Þ x ; 2 x 2

(16)

vð1Þ ðx; y; zÞ ¼ v0 ðx; yÞ 

h2 ð2Þ h1 ð1Þ f  fy þ zð1Þ fð1Þ y ; 2 y 2

(17)

wð1Þ ðx; y; zÞ ¼ w0 ðx; yÞ; where hk are the kth layer thickness z-coordinates. Deformations for layer k are given by 8 9 8 ðkÞ 9 > quðkÞ > > > qx xx > > > > > > > > > > > > ðkÞ > ðkÞ > > > > > qv > > > > > >  qy yy > > > > < = < = ðkÞ ðkÞ ðkÞ qu qv gxy ¼ : þ qy qx > > > > > > > > > > > > ðkÞ > > quðkÞ þ qwðkÞ > gxz > > > > > > qz > > qx > > > : gðkÞ > > ; > > > ðkÞ ðkÞ : qv þ qw > ; yz qz qy

(18) and

ðkÞ

z

Therefore in-plane deformations can be expressed as 8 ðkÞ 9 8 mðkÞ 9 8 f ðkÞ 9 8 mf ðkÞ 9 > > < xx > = > < xx > = < xx > = > < xx > = ðkÞ mðkÞ f ðkÞ ðkÞ ðkÞ yy yy ¼ yy þz þ mf yy > > : ðkÞ > ; > : mðkÞ > ; : f ðkÞ > ; > : mf ðkÞ > ; gxy gxy gxy gxy and shear deformations as ( ) ( qw ) ðkÞ 0 þ f gðkÞ x qx xz ¼ qw : ðkÞ 0 gðkÞ yz qy þ fy The membrane components are given by 8 mðkÞ 9 8 qu0 9 > qx > > > < xx > = > < = qv0 mðkÞ yy ¼ : qy > > : mðkÞ > ; > > > : qu0 qv0 ; gxy qy þ qx

2 ½hk =2; hk =2

are

the

kth

layer

(19)

(20)

(21)

(22)

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The bending components can be expressed as 8 9 ðkÞ qfx 8 f ðkÞ 9 > > > > qx > > > > < xx > = > < = ðkÞ qfy f ðkÞ yy ¼ qy > > : f ðkÞ > ; > > > > ðkÞ ðkÞ > > gxy qfy > : qfx ; qy þ qx

(23)

and the membrane-bending coupling components for layers 2, 3 and 1, are, respectively, given as 8 mf ð2Þ 9 8 9 > < xx > = > <0> = mf ð2Þ yy ¼ 0 ; (24) > : mf ð2Þ > ; > : > ; 0 gxy 8 9 ð2Þ ð3Þ h3 qfx h2 qfx > 8 mf ð3Þ 9 > > > > > 2 qx þ 2 qx > >  > > > > xx ð2Þ ð3Þ < = < = qf qf h h y y mf ð3Þ 3 2 þ yy ¼ ; 2 qy 2 qy >



> ; > : mf ð3Þ > > > ð2Þ ð3Þ > > ð2Þ ð3Þ > > qf qf gxy > : h22 qfqyx þ qxy þ h23 qfqyx þ qxy > ; 8 9 ð2Þ ð1Þ h2 qfx h1 qfx > 8 mf ð1Þ 9 >   > > > > 2 qx 2 qx > >  > > > > xx ð2Þ ð1Þ < = < = qf qf h h y y mf ð1Þ 2 1   yy ¼ : 2 qy 2 qy >



> ; > : mf ð1Þ > > > ð2Þ ð1Þ > > ð2Þ ð1Þ > > qf qf gxy > :  h22 qfqyx þ qxy  h21 qfqyx þ qxy > ;

(25)

(26)

Neglecting sðkÞ z for each orthotropic layer, the stress–strain relations in the fiber local coordinate system can be expressed as 8 ðkÞ 9 2 3ðkÞ 8 ðkÞ 9 s1 > > > Q 0 0 0 Q 1 > > > > > 11 12 > > > > > > > ðkÞ ðkÞ > > > > > 7 6 > > > > s  Q Q 0 0 0 > > > 7 6 22 < 2 = 6 12 < 2 > = 7 ðkÞ 7 6 0 0 Q 0 0 ¼ (27) tðkÞ g 33 7 > 12 >; 6 12 > > > > > > 7 6 > > > > 0 0 Q44 0 5 > > > tðkÞ > gðkÞ > > > 4 0 23 > > > > > 23 > > > > : : ; ðkÞ ðkÞ ; 0 0 0 0 Q 55 t31 g31 where subscripts 1 and 2 are, respectively, the fiber and the normal to fiber in-plane directions, 3 is the direction normal to the plate, and the reduced stiffness components, QðkÞ ij ; are given by QðkÞ 11 ¼

E ðkÞ 1

; ðkÞ 1  nðkÞ ; n 12 21

QðkÞ 22 ¼

E ðkÞ 2 1

ðkÞ nðkÞ 12 n21

;

ðkÞ ðkÞ QðkÞ 12 ¼ n12 Q11 ;

ARTICLE IN PRESS A.J.M. Ferreira / International Journal of Mechanical Sciences 46 (2004) 1549–1569 ðkÞ QðkÞ 33 ¼ G 12 ; ðkÞ nðkÞ 21 ¼ n12

ðkÞ QðkÞ 44 ¼ G 23 ;

1555

ðkÞ QðkÞ 55 ¼ G 31 ;

E ðkÞ 2 E ðkÞ 1

ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ in which E ðkÞ 1 ; E 2 ; n12 ; G 12 ; G 23 and G 31 are material properties of lamina k: By performing adequate coordinate transformation, the stress–strain relations in the global x–y–z coordinate system can be obtained as 38 8 ðkÞ 9 2 ðkÞ 9 ðkÞ ðkÞ Q11 Q12 Q16 0 0 > ðkÞ s > > > xx xx > > > 6 > 7> > > > > > ðkÞ ðkÞ ðkÞ ðkÞ > ðkÞ > > > > > 7 6 s  > > > Q Q Q 0 0 > yy = > yy > > 6 12 > 7< 22 26 < = 7 6 ðkÞ ðkÞ ðkÞ ðkÞ 7 6 tðkÞ g ¼ 6 Q16 Q26 Q66 (28) 0 0 7> xy >: xy > > > > > 7> 6 > > > > ðkÞ ðkÞ ðkÞ ðkÞ > > gyz > 6 tyz > > > > Q44 Q45 7 0 0 > > > 4 0 > 5> > > : ðkÞ > : ðkÞ > ; ; ðkÞ ðkÞ tzx gzx 0 0 0 Q45 Q55

By considering y as the angle between x-axis and 1-axis, with 1-axis being the ðkÞ

first principal material axis, connected usually with fiber direction, the components Qij given as [13]

are

Q11 ¼ Q11 cos4 y þ 2ðQ12 þ 2Q66 Þsin2 y cos2 y þ Q22 sin4 y;

(29)

Q12 ¼ ðQ11 þ Q22  4Q66 Þsin2 y cos2 y þ Q12 ðsin4 y þ cos4 yÞ;

(30)

Q22 ¼ Q11 sin4 y þ 2ðQ12 þ 2Q66 Þsin2 y cos2 y þ Q22 cos4 y;

(31)

Q66 ¼ ðQ11 þ Q22  2Q12  2Q66 Þsin2 y cos2 y þ Q66 ðsin4 y þ cos4 yÞ;

(32)

Q44 ¼ Q44 cos2 y þ Q55 sin2 y;

(33)

Q55 ¼ Q55 cos2 y þ Q44 sin2 y:

(34)

As in higher-order theories, this layerwise theory does not need the use of shearcorrection factors as in first-order theories [12,42]. The equations of motion of this layerwise theory are derived from the principle of virtual displacements. In the present work, only symmetric laminates are considered, therefore u0 ; v0 and related stress resultants can be discarded.

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The virtual strain energy ðdUÞ; the virtual work done by applied forces ðdV Þ; assuming a threelayer laminate, are given by (Z " Z X 3 hk =2 f ðkÞ ðkÞ f ðkÞ mf ðkÞ sxx ðzdxx þ dmf Þ dU ¼ xx Þ þ syy ðzdyy þ dyy O0 k¼1

hk =2

#

)

f ðkÞ ðkÞ ðkÞ þ dgmf Þ þ txz dgðkÞ þ txy ðzdgxy xy xz þ tyz dgyz dz dx dy

Z

3  X

¼

O0 k¼1

mf ðkÞ f ðkÞ ðkÞ mf ðkÞ f ðkÞ ðkÞ mf ðkÞ N ðkÞ þ M ðkÞ þ M ðkÞ xx dxx xx dxx þ N yy dyy yy dyy þ N xy dgxy

 ðkÞ ðkÞ ðkÞ ðkÞ f ðkÞ dg þ Q dg þ Q dg þ M ðkÞ xy xy xz yz dx dy x y and

ð35Þ

Z qdw0 dx dy;

dV ¼ 

(36)

O0

where O0 denotes the mid-plane of the laminate, q is the external distributed load and 8 9   < N ðkÞ = Z hk =2 ab ðkÞ 1 ¼ dzk ; s ab : M ðkÞ ; z hk =2 ab QðkÞ a ¼

Z

hk =2 hk =2

tðkÞ az dzk ;

(37)

(38)

where a; b take the symbols x; y: Substituting for dU; dV ; into the virtual work statement, noting that the virtual strains can be expressed in terms of the generalized displacements, integrating by parts to relieve from any derivatives of the generalized displacements and using the fundamental lemma of the calculus of variations, we obtain the Euler–Lagrange equations [13] with respect to 7 degrees of freedom ð1Þ ð2Þ ð2Þ ð3Þ ð3Þ (w0 ; fð1Þ x ; fy ; fx ; fy ; fx ; fy ) (see Fig. 1): ! 3 X qQðkÞ qQðkÞ y x þ  q ¼ 0; (39) dw0 : qx qy k¼1 ð1Þ ð1Þ h1 qN ð1Þ qM ð1Þ h1 qN xy qM xy xx xx :  þ  þ Qð1Þ x ¼ 0; 2 qx qx 2 qy qy

(40)

dfð1Þ y :

ð1Þ ð1Þ ð1Þ ð1Þ h1 qN yy qM yy h1 qN xy qM xy  þ  þ Qð1Þ y ¼ 0; 2 qy qy 2 qx qx

(41)

dfð2Þ x :

ð1Þ ð3Þ ð2Þ h2 qN ð1Þ h2 qN ð3Þ qM ð2Þ h2 qN xy h2 qN xy qM xy xx xx xx   þ   þ Qð2Þ x ¼ 0; 2 qx 2 qx qx 2 qy 2 qy qy

(42)

dfð1Þ x

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dfð2Þ y :

ð1Þ ð3Þ ð2Þ ð1Þ ð3Þ ð2Þ h2 qN yy h2 qN yy qM yy h2 qN xy h2 qN xy qM xy   þ   þ Qð2Þ y ¼ 0; 2 qy 2 qy qy 2 qx 2 qx qx

1557

(43)

dfð3Þ x :

ð3Þ ð3Þ h3 qN ð3Þ qM ð3Þ h3 qN xy qM xy xx xx    þ Qð3Þ x ¼ 0; 2 qx qx 2 qy qy

(44)

dfð3Þ y :

ð3Þ ð3Þ ð3Þ ð3Þ h3 qN yy qM yy h3 qN xy qM xy    þ Qð3Þ y ¼ 0: 2 qy qy 2 qx qx

(45)

The Euler–Lagrange equations can be written in terms of the displacements by substituting strains and stress resultants into this equation as ! !! ðkÞ 3 2 2 X qf qfðkÞ q w ðkÞ q w0 ðkÞ y 0 þ Q44  q ¼ 0; (46) hk Q55 þ x þ dw0 : 2 2 qx qx qy qy k¼1 dfð1Þ x : 2 ð2Þ 2 ð1Þ h2 ð1Þ h2 q fx ð1Þ h1 q fx þ Q  1 Q11 11 2 2 qx2 2 qx2



!

2 ð2Þ 2 ð1Þ h2 ð1Þ h2 q fy h1 q fy  1 Q12 þ 2 2 qxqy 2 qxqy

!

2 ð1Þ h31 ð1Þ q2 fð1Þ h31 ð1Þ q fy x Q11  Q 12 qx2 12 12 qxqy

! !! 2 ð2Þ 2 ð1Þ 2 ð1Þ q f q f h21 ð1Þ h2 q2 fð2Þ h q f y y 1 x x  þ Q33  þ þ 2 2 qy2 qxqy 2 qy2 qxqy !

 2 ð1Þ q fy h3 ð1Þ q2 fð1Þ ð1Þ qw0 x  1 Q33 þ fð1Þ þ h1 Q55 þ ¼ 0; x 2 12 qy qxqy qx

ð47Þ

dfð1Þ y :

! ! 2 ð2Þ 2 ð1Þ 2 ð1Þ 2 q f q f h21 ð1Þ h2 q2 fð2Þ h q f h h h ð1Þ y y 1 2 1 x x  1 Q22  Q12 þ þ 2 2 qxqy 2 qxqy 2 2 qy2 2 qy2 ! ! 2 ð1Þ 2 ð1Þ q2 fð1Þ h31 h31 ð1Þ q2 fð1Þ ð1Þ q fx ð1Þ q fy y x   Q33 Q12 þ Q22 þ 12 qxqy qy2 12 qxqy qx2 ! !! q2 fð2Þ q2 fð1Þ h21 ð1Þ h2 q2 fð2Þ h1 q2 fð1Þ y y x x  Q33 þ þ þ 2 2 qxqy qx2 2 qxqy qx2

 ð1Þ qw0 þ h1 Q44 þ fð1Þ ¼ 0; y qy

ð48Þ

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dfð2Þ x :

! ! 2 ð2Þ 2 2 ð1Þ 2 q2 fð1Þ q f h2 h2 q2 fð2Þ h q f h h h ð1Þ ð1Þ ð1Þ y y 2 2 x x  Q12 h1 Q11 h1 þ Q11 1 þ 1  2 2 qx2 2 qx2 2 2 qxqy 2 qxqy ! 2 ð2Þ 2 ð2Þ h3 ð2Þ q fx ð2Þ q fy  2 Q11 þ Q 12 12 qx2 qxqy ! ! 2 ð2Þ 2 ð3Þ 2 2 ð3Þ h2 h2 q2 fð2Þ h2 ð3Þ h2 q fy h23 q fy ð3Þ ð3Þ h3 q fx x þ  Q12 h3  Q11 h3 þ Q11 2 2 qx2 2 qx2 2 2 qxqy 2 qxqy ! ! ! q2 fð2Þ q2 fð1Þ h2 ð1Þ h2 q2 fð2Þ h1 q2 fð1Þ y y x x þ h1  Q  þ þ 2 33 2 qy2 qxqy 2 qy2 qxqy ! 2 ð2Þ q f h32 ð2Þ q2 fð2Þ y x  Q þ 12 33 qy2 qxqy ! ! 2 ð2Þ 2 ð3Þ 2 ð2Þ q f h2 h2 q fy h23 h2 ð3Þ q2 fð3Þ ð3Þ h2 q fx y x   þ Q þ h3 Q33 2 2 qy2 2 qxqy 2 2 33 qy2 qxqy

 ð2Þ qw0 ð2Þ þ h2 Q55 þ fx ¼ 0; qx

dfð2Þ y : 2 2 ð1Þ h2 h2 q2 fð2Þ ð1Þ ð1Þ h q fx x  Q12 h1 þ Q12 1 2 2 qxqy 2 qxqy



!

2 ð2Þ 2 ð1Þ h2 ð1Þ h2 q fy h21 q fy  Q22 h1 þ 2 2 qy2 2 qy2

ð49Þ

!

2 ð2Þ h32 ð2Þ q2 fð2Þ h3 ð2Þ q fy x Q12  2 Q22 12 qxqy 12 qy2

! ! 2 ð2Þ 2 ð3Þ 2 2 ð3Þ h2 h2 q2 fð2Þ h2 ð3Þ h2 q fy h23 q fy ð3Þ ð3Þ h3 q fx x  Q22 h3  Q12 h3 þ þ Q12 2 2 qxqy 2 qxqy 2 2 qy2 2 qy2 ! !! q2 fð2Þ q2 fð1Þ h2 ð1Þ h2 q2 fð2Þ h1 q2 fð1Þ y y x x þ þ  þ h1 Q  2 33 2 qxqy qx2 2 qxqy qx2 ! 2 ð2Þ q f h32 ð2Þ q2 fð2Þ y x þ  Q 12 33 qxqy qx2 ! ! 2 ð2Þ 2 ð3Þ 2 ð2Þ q f h2 h 2 q fy h33 h2 ð3Þ q2 fð3Þ ð3Þ h2 q fx y x h3 Q33 þ þ   Q 2 2 qxqy 2 qx2 2 2 33 qxqy qx2

 ð2Þ qw0 ð2Þ þ fy ¼ 0; þ h2 Q44 qy

ð50Þ

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dfð3Þ x : 2 ð2Þ 2 ð3Þ h2 ð3Þ h2 q fx ð3Þ h3 q fx þ Q  3 Q11 11 2 2 qx2 2 qx2

!

2 ð2Þ 2 ð3Þ h23 ð3Þ h2 q fy h3 q fy  Q12 þ 2 2 qxqy 2 qxqy

1559

!

2 ð3Þ h33 ð3Þ q2 fð3Þ h33 ð3Þ q fy x Q11  Q 12 qx2 12 12 qxqy ! !! q2 fð2Þ q2 fð3Þ h23 ð3Þ h2 q2 fð2Þ h3 q2 fð3Þ y y x x  þ Q  þ þ 2 33 2 qy2 qxqy 2 qy2 qxqy !

 2 ð3Þ h33 ð3Þ q2 fx ð3Þ q fy ð3Þ qw0 ð3Þ þ h3 Q55  Q þ þ fx ¼ 0; qy2 12 33 qxqy qx



dfð3Þ y :

!

2 ð2Þ 2 ð3Þ h2 ð3Þ h2 q fy h3 q fy  3 Q22 þ 2 2 qy2 2 qy2 ! 2 ð3Þ q2 fð3Þ h33 ð3Þ q2 fð3Þ h33 ð3Þ q fy h33 ð3Þ q2 fð3Þ y x x   þ Q Q  Q 12 12 qxqy 12 22 qy2 12 33 qyqy qx2 ! !! 2 ð2Þ 2 ð3Þ 2 ð3Þ q f q f h23 ð3Þ h2 q2 fð2Þ h q f y y 3 x x  Q  þ þ þ 2 33 2 qxqy qx2 2 qxqy qx2

 ð3Þ qw0 ¼ 0: þ h3 Q44 þ fð3Þ y qy

2 ð2Þ 2 ð3Þ h2 ð3Þ h2 q fx ð3Þ h3 q fx þ Q12  3 Q12 2 2 qxqy 2 qxqy

ð51Þ

!

ð52Þ

4. Interpolation for differential governing equations Applying the thin-plate splines method previously explained, the governing differential equations (46)–(52) are now interpolated, for each node i: Eq. (46) is then expressed as ! 2 3 NN NN X X ðkÞ ðkÞ X w q gj fx qgj dw0 : hk Q55 aj þ aj qx2 qx j¼1 j¼1 k¼1 !! 2 NN NN ðkÞ X fy qgj ðkÞ X w q gj þ Q44 aj þ a  q ¼ 0; ð53Þ j qy2 qy j¼1 j¼1 where gj was defined in Eq. (1) and NN represents the total number of discretization points. The other 6 equations are interpolated in a similar way. The vector of unknowns ð1Þ

f

ð1Þ

f

f

ð2Þ

ð2Þ

f

ð3Þ

f

ð3Þ

f

awj ; aj x ; aj y ; aj x ; aj y ; aj x ; aj y has dimension 7NN:

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5. Boundary conditions interpolation For each boundary node, the interpolation follows the approach in Eq. (4). As an example a simply supported condition in x ¼ b edge with outward normal direction a imposes seven boundary conditions, as follows: w0 ¼ 0;

(54)

M ðkÞ aa ¼ 0;

(55)

fðkÞ b ¼ 0:

(56)

These conditions are equivalent to w0 ¼ 0;

 h1 ð1Þ h1 ð1Þ ð1Þ ð1Þ dfx  N xx  N xy þ M xx 2 2

 h h2 ð3Þ h2 ð1Þ h2 ð3Þ 2 ð2Þ ð1Þ ð2Þ þ dfx  N xx þ N xx  N xy þ N xy þ M xx 2 2 2 2

 h3 ð3Þ ¼ 0; þ dfð3Þ N þ M ð3Þ xx x 2 xx fðkÞ b ¼ 0:

(57)

ð58Þ (59)

The interpolation of boundary equations leads to a change in the global equations system. For each node i were the equations are valid, the following multiquadric equations are imposed. For example, Eq. (57) is interpolated as NN X

awj gi ¼ 0;

(60)

j¼1

where NN represents the total number of grid points. The other boundary conditions are interpolated in the same way.

6. Numerical examples 6.1. Four layer ð0 =90 =90 =0 Þ square cross-ply laminated plate under sinusoidal load A square laminate of side a and thickness h is composed of four equally thick layers oriented at ð0 =90 =90 =0 Þ: It is simply supported on all edges and subjected to a sinusoidal vertical pressure of the form px py sin pz ¼ P sin a a with the origin of the coordinate system being located at the lower left corner on the mid-plane.

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The material properties are E 1 ¼ 25:0E 2 ;

G12 ¼ G13 ¼ 0:5E 2 ;

G 23 ¼ 0:2E 2 ;

n12 ¼ 0:25:

In this problem, in order to accommodate this three-layer formulation, layers at 90 are considered as one single layer. As will be seen further, this will not affect the solution. The problem was modeled with a regular node grid, in the domain nodes as well as in the boundary nodes. In Tables 1–3, a laminated composite plate is analyzed with N ¼ 11  11; 15  15 and 21  21 points. The numerical results are presented in a normalized form, as indicated by the following relations: 102 wða=2; a=2; 0Þh3 E 2 sxx ða=2; a=2; h=2Þh2 ; s ¼ ; xx Pa4 Pa2 txy ða; a; h=2Þh2 tzx ð0; a=2; firstlayerÞh ¼ : ; txy ¼ Pa Pa2

w¼ tzx

syy ¼

syy ða=2; a=2; h=4Þh2 ; Pa2

Table 1 ð0 =90 =90 =0 Þ square laminated plate under sinusoidal load, ah ¼ 4 a h

Method

w

sx

sy

tzx

txy

4

3 strip [49] HSDT [14] FSDT [50] Elasticity [54] Multiquadrics [51] ðN ¼ 11Þ Multiquadrics [51] ðN ¼ 15Þ Multiquadrics [51] ðN ¼ 21Þ Present, layerwise ðN ¼ 11Þ Present, layerwise ðN ¼ 15Þ Present, layerwise ðN ¼ 21Þ

1.8939 1.8937 1.7100 1.954 1.8804 1.8846 1.8864 1.8990 1.9023 1.9056

0.6806 0.6651 0.4059 0.720 0.6665 0.6660 0.6659 0.6505 0.6396 0.6420

0.6463 0.6322 0.5765 0.666 0.6292 0.6307 0.6313 0.6227 0.6236 0.6257

0.2109 0.2064 0.1398 0.270 0.1415 0.1372 0.1352 0.2078 0.2147 0.2160

0.0450 0.0440 0.0308 0.0467 0.0423 0.0429 0.0433 0.0418 0.0429 0.0437

Table 2 ð0 =90 =90 =0 Þ square laminated plate under sinusoidal load, ah ¼ 10 a h

Method

w

sx

sy

tzx

txy

10

3 strip [49] HSDT [14] FSDT [50] Elasticity [54] Multiquadrics [51] ðN ¼ 11Þ Multiquadrics [51] ðN ¼ 15Þ Multiquadrics [51] ðN ¼ 21Þ Present, layerwise ðN ¼ 11Þ Present, layerwise ðN ¼ 15Þ Present, layerwise ðN ¼ 21Þ

0.7149 0.7147 0.6628 0.743 0.7142 0.7150 0.7153 0.7250 0.7277 0.7298

0.5589 0.5456 0.4989 0.559 0.5464 0.5465 0.5466 0.5434 0.5466 0.5485

0.3974 0.3888 0.3615 0.403 0.4380 0.4382 0.4383 0.3930 0.3942 0.3951

0.2697 0.2640 0.1667 0.301 0.3267 0.3305 0.3347 0.2888 0.2950 0.2980

0.0273 0.0268 0.0241 0.0276 0.0264 0.0266 0.0267 0.0267 0.0269 0.0271

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Table 3 ð0 =90 =90 =0 Þ square laminated plate under sinusoidal load, ah ¼ 100 a h

Method

w

sx

sy

tzx

txy

100

3 strip [49] HSDT [14] FSDT [50] Elasticity [54] Multiquadrics [51] ðN ¼ 11Þ Multiquadrics [51] ðN ¼ 15Þ Multiquadrics [51] ðN ¼ 21Þ Present, layerwise ðN ¼ 11Þ Present, layerwise ðN ¼ 15Þ Present, layerwise ðN ¼ 21Þ

0.4343 0.4343 0.4337 0.4347 0.4535 0.4406 0.4365 0.5655 0.4659 0.4430

0.5507 0.5387 0.5382 0.539 0.5596 0.5456 0.5413 0.6873 0.5740 0.5484

0.2769 0.2708 0.2705 0.271 0.3427 0.3376 0.3359 0.3016 0.3434 0.2706

0.2948 0.2897 0.1780 0.339 0.4417 0.4223 0.4106 0.0434 0.2765 0.2911

0.0217 0.0213 0.0213 0.0214 0.0229 0.0218 0.0215 0.0362 0.0256 0.0226

It can be seen that the present layerwise approach with polyharmonic splines interpolation is very accurate for the analysis of composite laminates. Both the transverse displacement, normal and transverse stresses are very accurately predicted, for all ratios a=h: In the case of thick-plates (small a=h) the solution is better than other first-order and higher-order shear deformation approaches [49,14,50]. In other cases although our results are better than the FSDT and HSDT, the difference is not so large. In fact in thin-plates a=h ¼ 100 the other approaches are perhaps a better choice. The present layerwise formulation is better than the third-order formulation presented by Ferreira et al. [51], particularly in thicker composite plates. Unlike the FSDT and HSDT methods, and previous work by the author [12,42,51] where transverse shear stresses are calculated by the equilibrium equations, in the present formulation such transverse shear stresses are directly calculated by the constitutive laws. As can be seen in this and the next examples transverse shear stresses at each layer mid-surface are very accurately predicted. So this layerwise formulation combines the absence of shear-correction factors with direct calculation of transverse shear deformations, being this a significant advantage over laminate-wise FSDT and HSDT. 6.2. Three-layer square sandwich plate, under uniform load A simply supported sandwich square plate, under a uniform transverse load is considered. This is the classical sandwich example of Srinivas [52]. The material properties of the sandwich core are expressed in the stiffness matrix, Qcore as 3 2 0:999781 0:231192 0 0 0 7 6 0 0 0 7 6 0:231192 0:524886 7 6 7: 6 0 0 0:262931 0 0 Qcore ¼ 6 7 7 6 0 0 0 0:266810 0 5 4 0 0 0 0 0:159914

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Skins material properties are related with core properties by a factor R as Qskin ¼ RQcore : Transverse displacement and stresses are normalized through factors 0:999781 ; hq sð1Þ ða=2; a=2; h=2Þ 2 sð1Þ ða=2; a=2; 2h=5Þ 3 sð2Þ ða=2; a=2; 2h=5Þ ; sx ¼ x ; sx ¼ x ; s1x ¼ x q q q sð1Þ sð1Þ sð2Þ y ða=2; a=2; h=2Þ y ða=2; a=2; 2h=5Þ y ða=2; a=2; 2h=5Þ ; s2y ¼ ; s3y ¼ ; s1y ¼ q q q tð2Þ ð0; a=2; 0Þ 2 tð2Þ ð0; a=2; 2h=5Þ t1xz ¼ xz ; txz ¼ xz : q q w ¼ wða=2; a=2; 0Þ

Transverse displacement and stresses for a sandwich plate are indicated in Tables 4–6, and compared with various formulations. These formulations provide very good results both for displacement and stresses. It can be seen that the present formulation achieves very good results for all cases, without the use of shear-correction factors. The FSDT and HSDT results of Pandya [53] are inferior in accuracy when compared to our formulation for sandwich laminates where skin properties are quite different than core properties, which is the typical industrial case. So for R ¼ 15 or larger the present formulation should be adopted. The work of the author in laminated shell finite elements [12] and multiquadrics [12] using a first-order shear deformation approach is also compared. The results are as good or better than the present formulation. However, this was achieved by a shear-correction procedure [12] that is dependent on some assumptions that may not be general, although quite good for all tested cases so far. The present layerwise formulation is better than the third-order formulation presented by Ferreira et al. [51], particularly in sandwich plates with skin properties are much higher than core properties. For sandwich structures this layerwise theory coupled with polyharmonic (thin-plate) splines represents a very accurate alternative to finite element formulations.

7. Conclusions In this paper the analysis of composite laminated plates by the use of thin-plate splines RBFs [44,29] and using a layerwise formulation is performed for the first time. The Euler–Lagrange equations were derived and interpolated. Boundary conditions interpolation was schematically formulated. The results obtained showed excellent accuracy for the composite and the sandwich plate cases. The present methodology was compared with first-order shear-deformation and multiquadrics interpolation proposed by the author in [12,51] and proved to be even better for the analysis of sandwich laminates.

1564 w s1x s2x s3x s1y s2y s3y t1xz t2xz (a/2,a/2,0) (a/2,a/2,h/2) (a/2,a/2,2h/5) (a/2,a/2,2h/5) (a/2,a/2,h/2) (a/2,a/2,2h/5) (a/2,a/2,2h/5) (0,a/2,0) (0,a/2,2h/5)

5 HOST [53] FOST [53] CLT Ferreira [12] FerreiraðN ¼ 15Þ [12] Exact [52] Multiquadrics [51] ðN ¼ 11Þ Multiquadrics [51] ðN ¼ 15Þ Multiquadrics [51] ðN ¼ 21Þ Present, layerwise ðN ¼ 11Þ Present, layerwise ðN ¼ 15Þ Present, layerwise ðN ¼ 21Þ

256.13 236.10 216.94 258.74 257.38 258.97 253.6710 256.2387 257.1100 250.6251 253.6063 256.0643

62.38 61.87 61.141 59.21 58.725 60.353 59.6447 60.1834 60.3660 58.5692 59.1688 59.6661

46.91 49.50 48.623 45.61 46.980 46.623 46.4292 46.8581 47.0028 45.1931 45.6479 46.0467

9.382 9.899 9.783 9.122 9.396 9.340 9.2858 9.3716 9.4006 9.0386 9.1296 9.2093

38.93 36.65 36.622 37.88 37.643 38.491 38.0694 38.3592 38.4563 37.5045 37.8698 38.1519

30.33 29.32 29.297 29.59 27.714 30.097 29.9313 30.1642 30.2420 29.3324 29.6106 29.8373

6.065 5.864 5.860 5.918 4.906 6.161 5.9863 6.0328 6.0484 5.8665 5.9221 5.9675

3.089 3.313 4.5899 3.593 3.848 4.3641 3.8449 4.2768 4.5481 3.6864 3.9355 4.0304

2.566 2.444 3.386 3.593 2.839 3.2675 1.9650 2.2227 2.3910 1.7531 2.5949 2.5818

ARTICLE IN PRESS

R Method

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Table 4 Square laminated plate under uniform load

R Method

10 HOST [53] 152.33 FOST [53] 131.095 CLT 118.87 Ferreira [12] 159.402 Ferreira ðN ¼ 15Þ [12] 158.55 Exact [52] 159.38 Multiquadrics [51] ðN ¼ 11Þ 153.0084 Multiquadrics [51] ðN ¼ 15Þ 154.2490 Multiquadrics [51] ðN ¼ 21Þ 154.6581 Present, layerwise ðN ¼ 11Þ 153.8486 Present, layerwise ðN ¼ 15Þ 155.9313 Present, layerwise ðN ¼ 21Þ 157.5186

64.65 67.80 65.332 64.16 62.723 65.332 64.7415 65.2223 65.3809 63.1036 63.9061 64.5054

51.31 54.24 48.857 47.72 50.16 48.857 49.4716 49.8488 49.9729 47.0605 47.6845 48.1692

5.131 4.424 5.356 4.772 5.01 4.903 4.9472 4.9849 4.9973 4.7061 4.7685 4.8169

42.83 40.10 40.099 42.970 42.565 43.566 42.8860 43.1521 43.2401 42.3933 42.8637 43.2086

33.97 32.08 32.079 42.900 34.052 33.413 33.3524 33.5663 33.6366 32.5078 32.8734 33.1535

3.397 3.208 3.208 3.290 3.400 3.500 3.3352 3.3566 3.3637 3.2508 3.2873 3.3154

3.147 3.152 4.3666 3.518 3.596 4.0959 2.7780 3.1925 3.5280 3.6198 3.8018 3.8938

2.587 2.676 3.7075 3.518 3.053 3.5154 1.8207 2.1360 2.3984 2.6581 3.4712 3.2978

ARTICLE IN PRESS

w s1x s2x s3x s1y s2y s3y t1xz t2xz (a/2,a/2,0) (a/2,a/2,h/2) (a/2,a/2,2h/5) (a/2,a/2,2h/5) (a/2,a/2,h/2) (a/2,a/2,2h/5) (a/2,a/2,2h/5) (0,a/2,0) (0,a/2,2h/5)

A.J.M. Ferreira / International Journal of Mechanical Sciences 46 (2004) 1549–1569

Table 5 Square laminated plate under uniform load

1565

1566

R Method

15 HOST [53] 110.43 FOST [53] 90.85 CLT 81.768 Ferreira [12] 121.821 Ferreira ðN ¼ 15Þ [12] 121.184 Exact [52] 121.72 Multiquadrics [51] ðN ¼ 11Þ 113.5941 Multiquadrics [51] ðN ¼ 15Þ 114.3874 Multiquadrics [51] ðN ¼ 21Þ 114.6442 Present, layerwise ðN ¼ 11Þ 117.3458 Present, layerwise ðN ¼ 15Þ 118.9852 present, layerwise ðN ¼ 21Þ 120.2259

66.62 70.04 69.135 65.650 63.214 66.787 66.3646 66.7830 66.9196 64.3368 65.2210 65.8722

51.97 56.03 55.308 47.09 50.571 48.299 49.8957 50.2175 50.3230 46.3064 47.0207 47.5476

3.465 3.753 3.687 3.140 3.371 3.238 3.3264 3.3478 3.3549 3.0871 3.1347 3.1698

44.92 41.39 41.410 45.850 45.055 46.424 45.2979 45.5427 45.6229 45.1293 45.6486 46.0339

35.41 33.11 33.128 34.420 36.044 34.955 34.9096 35.1057 35.1696 33.9462 34.3643 34.6785

2.361 2.208 2.209 2.294 2.400 2.494 2.3273 2.3404 2.3446 2.2631 2.2910 2.3119

3.035 3.091 4.2825 3.466 3.466 3.9638 2.1686 2.6115 3.0213 3.5442 3.7060 3.8015

2.691 2.764 3.8287 3.466 3.099 3.5768 1.5578 1.9271 2.2750 3.3096 4.0794 3.7770

ARTICLE IN PRESS

w s1x s2x s3x s1y s2y s3y t1xz t2xz (a/2,a/2,0) (a/2,a/2,h/2) (a/2,a/2,2h/5) (a/2,a/2,2h/5) (a/2,a/2,h/2) (a/2,a/2,2h/5) (a/2,a/2,2h/5) (0,a/2,0) (0,a/2,2h/5)

A.J.M. Ferreira / International Journal of Mechanical Sciences 46 (2004) 1549–1569

Table 6 Square laminated plate under uniform load

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1567

Acknowledgements The Fundac- a˜o para a Cieˆncia e a Tecnologia has granted the author with various research contracts. Their support is gratefully acknowledged.

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