Nonlinear analysis of laminated composite plates using the mesh-free kp-Ritz method based on FSDT

Nonlinear analysis of laminated composite plates usingthe mesh-free kp-Ritz method based on FSDT

K.M. Liew a,b, J. Wang a, M.J. Tan b, S. Rajendran b,*

a Nanyang Centre for Supercomputing and Visualisation, Nanyang Technological University, Nanyang Avenue,

Singapore 639798, Singaporeb School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore

Received 4 September 2003; received in revised form 15 December 2003; accepted 23 March 2004

Abstract

A mesh-free kp-Ritz method of solution based on the kernel particle approximation for the field variables is pro-

posed for the large deflection flexural analysis of laminated composite plates. The first-order shear deformation theory

(FSDT) is used for modeling the flexure. The nonlinear solution algorithm is based on the total Lagrangian formulation

with Green’s strain measures and von Karman assumptions. The incremental form of nonlinear equations is obtained

by Taylor series expansion, and Newton’s method is used to solve these equations. Test problems involving square and

rectangular composite plates with SSSS and CCCC boundary conditions are solved to assess the efficacy of the pro-

posed method. The results are in excellent agreement with the series solution as well as the finite element solution

already reported in the literature.

2004 Published by Elsevier B.V.

Keywords: kp-Ritz method; Nonlinear bending; Laminated composite plates; FSDT; Newton methods; Arc length

1. Introduction

The nonlinear analysis of plates treats the stiffnesses and/or the loads as dependent upon the displace-

ments. Nonlinear problems can be classified as material nonlinear and geometric nonlinear problems [1].

The material nonlinearity is associated with the changes in material properties, e.g. elastoplastic problems.

The geometric nonlinearity is associated with significant changes in configuration (geometry). This paper

deals with only the geometric nonlinearity of composite laminated plates.Linear analyses may lead to inaccurate and inadmissible results under certain conditions and loadings.

In linear analysis of composite plates, the displacements and rotations are assumed to be small, and the

material property is regarded as linearly elastic. In the nonlinear analysis of laminated composite plates

*Corresponding author. Address: School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang

Avenue, Singapore 639798, Singapore. Tel.: +65-6790-5865; fax: +65-6791-1859.

E-mail address: [email protected] (S. Rajendran).

0045-7825/$ - see front matter 2004 Published by Elsevier B.V.

doi:10.1016/j.cma.2004.03.013

Comput. Methods Appl. Mech. Engrg. 193 (2004) 4763–4779

www.elsevier.com/locate/cma

mail to: [email protected]

based on a laminate theory with the von Karman strains, the geometry of the plates is assumed to beunchanged during the loading process. For problems with small strains and moderately large deflection and

rotations, these assumptions lead to very good results [2]. The first-order shear deformation plate theory is

used in this paper. In continuum mechanics, there are two incremental formulations to describe the

deformation and stress fields, i.e., Total Lagrangian (TL) formulation and Updated Lagrangian (UL)

formulation. In this paper, TL formulation is employed to describe the nonlinear behaviour of plates under

large deformations. Thus, all variables are referred to a fixed configuration, and changes in displacement

and stress fields are determined with respect to the fixed (reference) configuration.

Presently, the finite element method continues to dominate the numerical analyses of geometricalnonlinear problems. Many researchers have used it to solve large deformation problems of plates [3–7]. For

geometrically nonlinear problems, the incremental formulation is widely used. In the present study, the

discretization of composite laminated plates for nonlinear analysis is formulated in incremental form. The

formulations are derived by Ritz method with kernel particle discretization.

The numerical analysis of nonlinear problems leads to nonlinear algebraic equations. To solve these

equations, several methods have been developed. Among them, the Newton–Raphson method, modified

Newton–Raphson method and quasi-Newton method are widely used. Clarke and Hancock [7] have re-

viewed incremental-iterative solution techniques for geometrically nonlinear analyses. In this paper, themodified Newton–Raphson method is combined with the arc-length technique continuation to solve the

nonlinear equations.

In engineering analysis of nonlinear problems, the finite element method is still the most commonly used

method. Nevertheless, though relatively new, a class of particle methods or meshless methods has proven to

be a potentially attractive alternate numerical technique to solve the nonlinear problems especially for those

with large deformations [8–12]. The present work is an extension of our previous efforts to study the

laminated composite plates using the kernel particle Ritz method. The kernel particle Ritz (kp-Ritz)

method belongs to the broad class of meshless methods, and attempts a FSDT Ritz solution based onkernel particle approximations. In our previous works, the bending, free vibration and buckling analysis of

composite plates have been studied [13–15]. Free vibration analysis of laminated composite panels has also

been carried out [16–18]. In this paper, the nonlinear deformation of laminated composite plates is studied

using the kp-Ritz method.

2. Formulation for nonlinear flexure of plates

2.1. Total potential energy

The formulation for the large deflection analysis of laminated composite plates is derived here from Ritz

method based on the first-order shear deformation plate theory (FSDT). In geometrically nonlinear

analyses, the nonlinear terms are included in the strains.

In the first order plate shear deformation theory [2], the displacement field of a laminated composite

plate is expressed as

uðx; y; zÞ ¼ u0ðx; yÞ þ zhxðx; yÞ;vðx; y; zÞ ¼ v0ðx; yÞ þ zhyðx; yÞ;wðx; y; zÞ ¼ w0ðx; yÞ;

ð1Þ

where (u; v;w) are the displacements of a generic point (x; y; z) in the laminated plate, and (u0; v0;w0) are the

displacement projections on the mid-plane, and (hx; hy) denote the rotations of the transverse normal about

y- and x-axis, respectively. Considering Green’s expression as well as the lower order terms in a manner

4764 K.M. Liew et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) 4763–4779

consistent with the usual von Karman assumptions, which concerns small strains and moderately large

rotations, the strain components at a generic point of the plate are expressed as

e ¼

exeyczxcyzcxy

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼

exey2ezx2eyz2exy

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼

ouox

þ zohxox

þ 1

2

owox

2

ovoy

þ zohyoy

þ 1

2

owoy

2

owox

þ hx

owoy

þ hy

ouoy

þ ovox

þ zohxoy

þ ohyox

þ ow

oxowoy

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>;

; ð2Þ

which are re-expressed as linear and nonlinear components as

eL ¼eLxeLycLxy

8><>:9>=>; ¼

eLxeLy2eLxy

8><>:9>=>; ¼

ouoxovoy

ouoy

þ ovox

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;; ð3Þ

c ¼czxcyz

( )¼

2ezx2eyz

¼

owox

þ hx

owoy

þ hy

8>><>>:9>>=>>;; ð4Þ

j ¼jx

jy

jxy

8><>:9>=>; ¼

ohxoxohyoy

ohxoy

þ ohyox

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;ð5Þ

and

eNL ¼eNLx

eNLy

cNLxy

8><>:9>=>; ¼

eNLx

eNLy

2eNLxy

8><>:9>=>; ¼

1

2

owox

2

1

2

owoy

2

owox

owoy

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;: ð6Þ

With the usual assumptions of FSDT, the force and moment resultants of the plates are expressed asbNcMbQ8<:

9=; ¼A B 0

BT D 0

0 0 E

24 35 eL þ eNL

j

c

8<:9=; ð7Þ

K.M. Liew et al. / Comput. Methods Appl. Mech. Engrg. 193 (2004) 4763–4779 4765

in which, the membrane, membrane-bending, bending and transverse shear stiffness terms are repre-

sented by the sub-matrices, i.e., A, B, D and E, respectively. The strain energy of the laminated plate is

expressed as

U ¼ 1

2

Z ZX

eTL þ eTNL jT cT A B 0

BT D 0

0 0 E

24 35 eL þ eNL

j

c

8<:9=;dxdy

¼ 1

2

Z ZX

eTL þ eTNL jT A B

BT D

eL þ eNL

j

dxdy þ 1

2

Z ZXcTEcdxdy: ð8Þ

Considering a uniformly distributed transverse load q, the external work is given by

W ¼Z Z

Xwqdxdy: ð9Þ

Thus the total potential energy of the plate becomes

P ¼ U W

¼ 1

2

Z ZX

eTL þ eTNL jT A B

BT D

eL þ eNL

j

dxdy þ 1

2

Z ZXcTEcdxdy

Z ZXwqdxdy: ð10Þ

2.2. Ritz procedure for nonlinear equilibrium equations

By discretizing the problem domain X into NP scattered particles ðx1; x2; . . . ; xNPÞ, the displacement field

of a laminated plate using the kernel particle Ritz approach can be expressed as

½u; v;w; hx; hy T ¼XNP

I¼1

kðx; x xIÞ~/ðx xIÞ½uI ; vI ;wI ; hxI ; hyI T ¼XNP

I¼1

WIðx; yÞd I : ð11Þ

where NP is the total number of the particles which are used to discretize the domain, WIðx; yÞ are the shapefunctions, and d I ¼ ½uI ; vI ;wI ; hxI ; hyI T is a vector of unknown displacements of Ith particle. The shape

functions can be expressed by

WIðxÞ ¼ kðx; x xIÞ~/ðx xIÞ ¼ kðx; x xIÞ/ðx xIÞnðx xJ Þ

; ð12Þ

in which, kðx; x xIÞ is the coefficient function, /ðx xIÞ the kernel function and nðx xJ Þ the function to

introduce the singularity to the kernel function which is associated with the particle xI . The kernel function

and the singularity function are defined as

/ðx xIÞ ¼1

ax/

x xIax

1

ay/

y yIay

ð13Þ

and

nðx xIÞ ¼ nðx xI ; y yIÞ ¼x xIax

2"

þ y yIay

2#p

ð14Þ

in which, ax and ay are the kernel support measurement along x-axis and y-axis, respectively. p is the

singularity order (p > 0). The correction function kðx; x xIÞ is written as


kðx; x xIÞ ¼ aTðx xIÞbðxÞ; ð15Þ

where

aTðx xIÞ ¼ ½1; x xI ; y yI ; ðx xIÞ2; ðx xIÞðy yIÞ; ðy yIÞ2; ð16Þ

bTðxÞ ¼ ½b00ðxÞ; b10ðxÞ; b01ðxÞ; b20ðxÞ; b11ðxÞ; b02ðxÞ ð17Þ

and a is a vector of quadratic basis. The b(x) are functions of x which are yet to be determined. Thus, the

shape functions can be written as

WIðxÞ ¼ bTðxÞaðx xIÞ~/ðx xIÞ; ð18Þ

where

bðxÞ ¼ M1ðxÞað0Þ ð19Þ

and the moment matrix M is a function of x, given by

MðxÞ ¼XNP

I¼1

aðx xIÞaTðx xIÞ~/ðx xIÞ: ð20Þ

Therefore, the shape function can be expressed as

WIðxÞ ¼ aTð0ÞM1ðxÞaðx xIÞ~/ðx xIÞ: ð21Þ

In this paper, the cubic spline function

/ðzIÞ ¼

2

3 4z2I þ 4z3I for 06 zI 6 1

2

4

3 4zI þ 4z2I

4

3z3I for

1

26 zI 6 1

0 otherwise

8>>>><>>>>:

9>>>>=>>>>; ð22Þ

is chosen as the kernel function, where zI ¼ jx xI j=aI . The domain, in which /ðxJ xIÞ ¼ 1aI/ xJxI

aI

6¼ 0

for all particles xJ is called the support of /ðx xIÞ. The dilation parameter aI ¼ aðxIÞ is to control the size

of the support, and the parameter values are often different at various particles. In the numerical imple-

mentation, the dilation parameters are usually defined as aI ¼ sDI where s is a scaling factor, and DI is thelarger of the distances of the two neighbors of xI in one-dimensional case. For the 2-D problem, the dilation

parameters can similarly be defined in two directions.

With the assumptions of displacements as in Eq. (11), the strains of the composite laminate are expressed

using Eqs. (3)–(6) as

eL ¼XNP

I¼1

BLIdpI ; eNL ¼ 1

2

XNP

I¼1

Bndb; j ¼XNP

I¼1

BbIdbI ; c ¼XNP

I¼1

BqIdbI ; ð23Þ

where the displacements are re-expressed as two parts

dpI ¼ uI vI½ T; dbI ¼ wI hxI hyI½ T: ð24Þ


The strain matrices are given by

BLI ¼WI;x 0

0 WI ;y

WI;y WI ;x

24 35; BnI ¼

owox

0

0owoy

owoy

owox

26666664

37777775oWI

ox0 0

oWI

oy0 0

26643775; ð25Þ

BbI ¼0 WI;x 00 0 WI ;y

0 WI ;y WI ;x

24 35; BqI ¼WI ;x WI 0

WI;y 0 WI

: ð26Þ

The nonlinear equations of the plate can be derived by Ritz method. First, the strain energy is divided into

two parts UL and UN, corresponding to the linear strains and nonlinear strains, respectively. Expanding

Eq. (10), one obtains

U ¼ UL þ UN ¼ 1

2

Z ZX

eTLAeL

þ eTLBjþ jTBeL þ jTDjþ cTEcdxdy

þ 1

2

Z ZX

eTLAeNL

þ eTNLAeL þ eTNLBjþ jTBeNL þ jTDeNL þ eTNLAeNL

dxdy: ð27Þ

The Ritz minimization of the total potential energy yields the equations

oPodI

¼ oUL

odIþ oUN

odI oW

odI¼ 0; dI ¼ uI ; vI ;wI ; hxI ; hyI : ð28Þ

Using Eqs. (23) and (27),

oUL

odI¼ K0d: ð29Þ

where K0 is the linear stiffness matrix, a typical entry of which is defined as follows:

.TTT

TT

0 dxdyqJqIbJbILJbI

bJLILJLIIJ ∫∫Ω

+

=EBBDBBBBB

BBBABBK ð30Þ

From the nonlinear part of the strain energy

oUN

odI¼ Kn2d: ð31Þ

For the purpose of this derivation, the nonlinear strains in Eq. (6) are re-written as

eNL ¼ 1

2

owox

0

0owoy

owoy

owox

26666664

37777775owoxowoy

8>><>>:9>>=>>; ¼ 1

2cH bP: ð32Þ


Then the following displacement-dependent stiffness matrix can be obtained as

.

2

1

2

12

10

TTTTT

T

2 dxdy

nJnInJbIbJnILJnI

nJLI

IJn ∫∫Ω

++=

ABBBBBBBBABB

ABBK ð33Þ

From the differential of the external work, the force vector is obtained as

F I ¼oWodI

¼ dxdy: ð34Þ

where qI ¼ ½0; 0; q; 0; 0T, and the shape function matrix is defined as

WI ¼

WI 0 0 0 0

0 WI 0 0 0

0 0 WI 0 00 0 0 WI 0

0 0 0 0 WI

266664377775: ð35Þ

Thus the nonlinear equilibrium equations may be obtained as

K sðdÞd ¼ F; ð36Þ

where the secant stiffness matrix is defined as

K sðdÞ ¼ K0 þ Kn2ðdÞ: ð37Þ

3. Solution of the nonlinear equations

3.1. Incremental form of nonlinear equations

For solving the nonlinear system, incremental-iterative Newton-type solution methods should be used.

To get the incremental form, Eq. (36) is re-expressed as

gðdÞ ¼ K sd F ¼ 0: ð38Þ

The applied load is assumed to be proportional to a fixed load F0 as

F ¼ kF0: ð39ÞThus the nonlinear equilibrium equations become functions of displacements and the load scaling factor k.Hence, Eq. (38) may be re-written as

gðd; kÞ ¼ K sðdÞd kF0 ¼ 0: ð40ÞFor an equilibrium configuration near the old configuration,

gðd þ Dd; kþ DkÞ ¼ 0: ð41ÞApplying the Taylor series expansion to the above equation

gðd þ Dd; kþ DkÞ ¼ gðd; kÞ þ K tDd DkF0 ¼ 0; ð42Þ


where Dd is the displacement increment as the load changes from kF0 to ðkþ DkÞF0, and the following

substitution is implicit:

oUodI

¼ K sd; ð43Þ

o2UodIodJ

¼ K t: ð44Þ

The tangent stiffness matrix can be obtained as

K t ¼ K0 þ Kn þ KG; ð45Þin which K0 is the linear stiffness matrix as in Eq. (31), KG the geometrical stiffness matrix and Kn the

displacement-dependant stiffness matrix. The last two matrices can be expressed as

dxdynJnInJbIbJnILJnI

nJLInIJ ∫∫ ∫∫Ω

++

=ABBBBBBBBABB

ABBK

TTTT

T0ð46Þ

and

ð47Þ

Finally the incremental form of the nonlinear equilibrium equations is expressed by using the tangent

stiffness matrix and the displacement increments as

K tDd ¼ DkF0 gðd; kÞ: ð48ÞThe incremental formulae for iteration are written as

Ddm ¼ ½K tðdmÞ1½DkmF0 gðdm; kmÞ¼ ½K tðdmÞ1½DkmF0 K sðdmÞdm þ kmF0;

dmþ1 ¼ dm þ Ddm;

8<: ð49Þ

where the subscript m refers to the load step number.

As Dk is a variable to be solved for each incremental step, an additional constraint equation is needed.

The arc-length continuation used here provides this constraint. In the arc-length continuation method, for

each Dk step, a corresponding external load increment is applied, and subsequently iterations are necessary

to reach a new equilibrium using Eq. (48). Let the superscript n denote the nth iteration cycle and subscript

m represents the mth load step, the equilibrium iterations start from n ¼ 2, and n ¼ 1 corresponds to theexternal load increment of every load step m. Then the generalized equation of the incremental-iterative

equation is

½K tmDdnm ¼ DknmF0 gðdn1

m ; kn1m Þ ¼ DknmF0 ½K sðdn1

m Þdm kn1m F0: ð50Þ

The displacement increment Dd is composed of two parts, one is from the external load increment and the

other is from the residual forces, i.e.

Ddnm ¼ ½ðK tÞm

1 DknmF0

½K sðdn1

m Þdm kn1m F0

¼ Dknm½d f m þ ½DdRnm; ð51Þ

where

½d f m ¼ ½ðK tÞm1F0 ð52Þ


and

½DdRnm ¼ ½ðK tÞm1gn1m ¼ ½ðK tÞm

1½K sðdn1m Þdn1

m kn1m F0: ð53Þ

The increment of the load level parameter Dknm is obtained by solving the arc-length constraint equation

(Eq. (58)). The total displacements and the updated load level parameter are obtained as

dnm ¼ dn1

m þ Ddnm; knm ¼ kn1

m þ Dknm: ð54Þ

3.2. Iterative strategy of the arc-length technique

In the modified Riks method proposed by Crisfield [19], the additional arc-length constraint equation is

written as

½ðDdaÞnmTðDdaÞnm ¼ l2m; ð55Þ

where lm is the generalized ‘arc-length’ of the tangent at converged state of the last load step in the load–

deflection space, and ðDdaÞnm are the accumulated displacements in the mth load step until the nth iteration,

i.e.

ðDdaÞnm ¼ dnm dm1: ð56Þ

Eq. (55) can be re-written as

½ðDdaÞn1

m þ Dknmðd fÞm þ ðDdRÞnmT½ðDdaÞn1

m þ Dknmðd fÞm þ ðDdRÞnm l2m ¼ 0; ð57Þ

and simplified into a quadratic equation,

a1ðDknmÞ2 þ a2Dk

nm þ a3 ¼ 0; ð58Þ

where the coefficients are defined as

a1 ¼ ðd fÞTmðd fÞm;a2 ¼ 2½ðDdaÞn1

m þ ðDdRÞnmTðd fÞm;

a3 ¼ ½ðDdaÞn1

m þ ðDdRÞnmT½ðDdaÞn1

m þ ðDdRÞnm l2m:

ð59Þ

The constraint equation may have two real roots, and the correct choice is the one makes the angle between

the incremental displacement vector of last iteration and that of current iteration positive. The angles aremeasured by

a1 ¼ ½ðDdaÞn1

m þ ðDknmÞ1ðd fÞm þ ðDdRÞnmTðDdaÞn1

m ;

a2 ¼ ½ðDdaÞn1

m þ ðDknmÞ2ðd fÞm þ ðDdRÞnmTðDdaÞn1

m :ð60Þ

The correct root is the one that makes a positive. If both are positive, the one that makes the a smaller is

selected.

3.3. Strategy for load incrementation

When the arc-length method is incorporated into the modified Newton method to solve the nonlinear

equation, the load incrementation is implicitly controlled. If the (m 1)th step load km1F0 and thecorresponding displacements dm1 have been known, then the arc-length in the mth load step is defined

as


l2m ¼ ðDk1mÞ2ðd fÞTmðd fÞ; ð61Þ

where Dk1m is the first load increment in the mth load step. Then the arc-length used in the current increment

can be obtained from the previous increment as

lm ¼ lm1

JdesJm1

0:5–1

; ð62Þ

where Jdes is the desired number of the iterations (3–5), and Jm1 is the number of iterations in the previous

(m 1)th load step. The new load increment parameter is obtained as

Dk1m ¼ lmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðd fÞTmðd fÞm

q : ð63Þ

To begin with, the first load increment Dk11, it is usually chosen to be 20–40% of the anticipated maximum

load.It should be noted that the load incremental parameter Dk1m may have different signs and the correct sign

is decided by Crisfield’s method [19]. When the determinant of the tangent stiffness matrix does not change

sign, the current sign of Dk1m follows that of the previous increment. Otherwise, Dk1m changes its sign.

4. Numerical examples

In this section, a few examples of composite laminated plates with large enough loads to cause significantgeometric nonlinearity are numerically investigated. Simply supported and clamped boundary conditions

are considered. The results are compared with those calculated by other methods. Exact or close-form

solutions of nonlinear analysis of plates are not available yet. Hence in this paper, the nonlinear deflections

of plates are compared with experimental results or numerical results of other methods such as the finite

element method.

In the following examples, regular particle distribution is set as 17 · 17. The Gauss integration scheme is

set as 2 · 2. Since the singularity has been introduced to the kernel function to construct the shape func-

tions, the boundary conditions can be treated as in FEM techniques. As no coordinate transformation isneeded, the singular kernel method is more suitable for nonlinear analysis than the other methods in

particle approximations. The coding procedures may be seen in Appendix A.

4.1. Isotropic plate

The nonlinear flexure of a square isotropic plate is carried out first. The plate is subjected to a uniformly

distributed transverse load of intensity q, and it is simply supported on all the four edges (SSSS). The

geometry and the load are shown as in Fig. 1. The geometry parameters and nondimensional materialproperties are set as

a ¼ b ¼ 10; h ¼ 0:2; E ¼ 3 106; v ¼ 0:316: ð64Þ

As the thickness-to-edge ratio is quite small (h=b ¼ 0:02), so the present results could be compared with

Chia’s results [20] for the thin plate.

Chia used series method to solve the nonlinear bending of isotropic plates with all simply supported

edges (SSSS) or clamped edges (CCCC). The deflection–load curve of the SSSS and CCCC isotropic plate


solved by the present method is shown in Figs. 2 and 3, respectively. The nondimensional deflection andloads in Fig. 2 are defined as the follows:

w ¼ wc

h; q ¼ qb4

Eh4; ð65Þ

where wc is the center deflection. Similarly, with respect to Fig. 3

w ¼ wc

h; q ¼ qb4

Dh; ð66Þ

where D ¼ Eh3=12ð1 m2Þ is the flexural rigidity of the plate.

The results are compared with the series solutions of Chia [20] which is based on thin plate theory.

Though a slight difference exists between the solutions, the agreement is still quite good.

0 8 16 24 32 40 48 56 64 72 80 880.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

w

q

Chia (1980) [20] Present

Fig. 2. The load–deflection curves of an SSSS square isotropic plate.

a

y

o b

x

qh

Fig. 1. An SSSS plate with uniformly distributed transverse loads.


4.2. Laminated composite plates

Symmetric and antisymmetric cross-ply composite laminated plates of square and rectangular geometry

are considered with fully clamped or simply supported edges. Uniformly distributed lateral pressures are

applied to the plates.First, a square four-layered symmetric cross-ply laminate (0/90/90/0) is considered. The geometry

and material properties are set as follows:

a ¼ b ¼ 12 in:; h ¼ 0:138 in:; E1 ¼ 1:8282 106 psi; E2 ¼ 1:8315 106 psi;

G12 ¼ G23 ¼ G13 ¼ 3:125 105 psi; m12 ¼ m23 ¼ m13 ¼ 0:23949: ð67Þ

The plate is clamped at the edges (CCCC), and thus the boundary conditions are set as

½u; v;w; hx; hy TI ¼ 0; ð68Þ

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.20.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

w

q

FEM (Ganapathi, 1996) [22] Experimental

(Zaghloul et al., 1975) [21] Present

Fig. 4. The load–deflection curves of a CCCC square symmetric cross-ply composite laminated plate ð0=90=90=0Þ.

0 25 50 75 100 125 150 175 200 225 250 275 300 3250.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

w

q

Chia (1980) [20] Present

Fig. 3. The load–deflection curves of a CCCC square isotropic plate.


where the subscript I refers to a typical node on the edges.

The plot of center deflection versus lateral load intensity is shown in Fig. 4. The present solution is

compared with available results obtained by experiments [21] and the finite element method [22]. The

present result is in close agreement with the finite element result, although it has a considerable deviation

from the experimental curve. This discrepancy is due to the assumption of an ideal clamped boundary

condition for the computation, which are difficult to ensure in experiments.

A rectangular three-layer symmetric cross-ply plate (0/90/0) is also studied. The nondimensional

material properties are as follows:

E1 ¼ 25E2; G12 ¼ G13 ¼ 0:5E2; G23 ¼ 0:2E2; m12 ¼ 0:25: ð69Þ

0 40 80 120 160 200 240 280 320 360 4000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

w

q

PresentSavithri et al. (1993) [23]

Fig. 5. The load–deflection curve of an SSSS square symmetric cross-ply composite laminated plate ½ð0=90=0Þ; h=b ¼ 0:1; a=b ¼ 1.

0 40 80 120 160 200 240 280 320 360 400 4400

25

50

75

100

125

150

175

200

225

250

275

σ x

q

Fig. 6. The load–stress curve of an SSSS square symmetric cross-ply composite laminated plate ½ð0=90=0Þ; h=b ¼ 0:1; a=b ¼ 1.


The plate is simply supported along all edges (SSSS), and is subjected to the uniformly distributed trans-

verse load. The geometric parameters h=b and a=b are varied from 0.01 to 0.1 and 0.5 to 2.0, respectively.

The results are compared with the series solutions of Savithri et al. [23]. For comparison, the nondimen-

sional load parameter, the center deflection and the extreme fibre stress are computed as:

q ¼ qb4

E2h4; w ¼ wc

h; rx ¼

rxb4

E2h4; ð70Þ

where q is the uniform lateral pressure, wc is the center transverse displacement of the plate and rx is the

extreme fibre stress. The load–displacement plots are shown in Figs. 5–9, which suggest that the results are

in close agreement with the series solutions.

0 40 80 120 160 200 240 280 320 360 4000.0

0.1

0.2

0.3

0.4

0.5

0.6

w

q

Present Savithri et al. (1993) [23]

Fig. 8. The load–deflection curve of an SSSS rectangular symmetric cross-ply composite laminated plate ½ð0=90=0Þ; h=b ¼0:1; a=b ¼ 0:5.

0 40 80 120 160 200 240 280 320 360 4000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

w

q

PresentSavithri et al. (1993) [23]

Fig. 7. The load–deflection curve of an SSSS rectangular symmetric cross-ply composite laminated plate ½ð0=90=0Þ; h=b ¼0:1; a=b ¼ 2.


5. Conclusions

In this paper, a kernel particle Ritz solution method has been proposed for the geometrically nonlinear

analysis of plate flexure, which is based on the total Lagrangian formulation and first-order shear defor-

mation plate theory. The nonlinear equations are solved by using the modified Newton’s method and arclength continuation. Typical problems involving isotropic as well as laminated composite plates have been

solved considering different aspect ratios and boundary conditions. The numerical results are in excellent

agreement with those obtained by finite element and series solution methods reported by previous

researchers. Thus the proposed kernel particle Ritz method promises to be potential alternative to the finite

element method in solving the large deformation plate flexure problems.

Appendix A. Coding procedure

For numerical implementation, the code includes three main steps, namely pre-processing, solving and

post-processing procedures. In the code developed for the present research, the nonlinear equations may be

solved by the modified Newton–Raphson method alone or combining it with the arc-length technique. The

algorithm is described in the following flow chart:

(1) Pre-processing step:

• Input the material properties and compute the stiffness coefficients A, B, D and E;• Input the fixed external loading, scaling factor s, and the geometry (length a, width b and thickness

h);• Set up particle coordinates;

• Determine the domain of influence of every particle;

• Read variables related with boundary conditions;

• Set up Gauss quadrature cells, Gauss points, weights, Jacobian;

• Compute the equivalent fixed external force vector F0 in Eq. (39);

0 40 80 120 160 200 240 280 320 360 4000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

w

q

a/b= 2a/b= 1a/b=0.5

Fig. 9. Comparisons of the load–deflection curves of SSSS rectangular symmetric cross-ply composite laminated plates with different

aspect ratios ½ð0=90=0Þ; h=b ¼ 0:1.


• Input the parameters for increment/iteration control;

For a new computation, set initial displacements, initial internal forces, initial load-level factor,

load factor increment, desired iteration times, d0, k0, Dk11, R0 and Jdes;

For a re-solution, read the displacements, internal forces, external forces and etc. of last successful

load step. d0, k0 and R0 from the previous saved file.

(2) Solution step:

• Start the load incremental loop m Save data of the (m 1)th load step; Compute the tangent stiffness K tðdm1Þ as in Eq. (45), check the sign of the determinant of K t to

determine if the limit point or singular point is passed;

Add boundary conditions and solve the tangent displacement ðd fÞm due to the fixed external force

F0: ðd fÞm ¼ ½ðK tÞm1F0;

Update the load level factor;

If m ¼ 1, compute first arc-length l21 ¼ ðDk11Þ2ðd fÞT1 ðd fÞ1;

If mP 2, compute arc length lm ¼ lm1

ffiffiffiffiffiffiffiJdesJm1

q;

Compute the initial load increment Dk1m ¼ lmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðd f ÞTmðdf Þm

p in the mth load step;

Update the total displacements and load level: d1m ¼ dm1 þ Dk1mðd fÞm, k

1m ¼ km1 þ Dk1m;

Save k1m; Start equilibrium iteration loop using the modified N–R method with arc-length continuation

method. In the nth iteration

– Calculate residual forces gn1m ¼ K sðdn1

m Þ kn1m F0;

– Save residual forces;

– Compute displacement increments: ðDdaÞn1

m ¼ dn1m dm1;

– Enforce boundary conditions, and compute ðDdRÞnm ¼ ½K t1

m gn1m ;

– Solve a1ðDknmÞ2 þ a2Dk

nm þ a3 ¼ 0 to obtain Dknm, and select correct root or stop the iteration if the

maximum load has been reached or there are no real roots;

– Update the load increment;

– Save increment of the load level and update the displacement and load level;

– Obtain the displacement increments: Ddnm ¼ Dknmðd fÞm þ ðDdRÞnm;

– Update total displacements: dnm ¼ dn1

m þ Ddnm;

– Update total load level: knm ¼ kn1m þ Dknm;

– Check convergence by using the norm of the residual forces: kgnmk6 bkFn

mk ¼ bk6 knmF0k;– If the norm is less than the tolerance limit, stop iteration and save the iteration number.

End of the iteration loop.

• End of the load increment loop.

(3) Post-processing step:

• Obtain the displacements and strain matrices in each load step, then calculate the stresses of the plate

at any point in the plate;

• Draw the deformation mode and stress distributions, and etc.

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Nonlinear analysis of laminated composite plates using the mesh-free kp-Ritz method based on FSDT