Analysis of laminated shear-flexible angle-ply plates

Composite Structures 67 (2005) 71–84

www.elsevier.com/locate/compstruct

Analysis of laminated shear-flexible angle-ply plates

Reaz A. Chaudhuri *

Department of Materials Science & Engineering, University of Utah, 1222 S. Central Campus Drive, Room 304, Salt Lake City, UT 84112-0560, USA

Available online 27 February 2004

Abstract

A recently developed C0-type triangular composite plate element, based on the assumption of transverse inextensibility and

layerwise constant shear-angle theory (LCST), is utilized to analyze antisymmetric and symmetric angle-ply plates subjected to

distributed transverse loading. Effect of numerical integration on the rate of convergence of displacements and moments is inves-

tigated in detail. Comparison of the numerical results computed using the present triangular element with their analytical coun-

terparts based on the classical lamination theory (CLT) in the thin plate regime also forms a major part of the present investigation.

Limited comparisons with the first-order shear deformation theory (FSDT) results, computed using assumed stress hybrid finite

elements, are also presented. Numerical results presented also include the effect of fiber orientation angle on the displacements and

moments for thin laminates. Finally, the effect of thickness on the computed transverse displacement (deflection), interfacial inplane

displacement and inplane stress is also thoroughly investigated.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Anisotropic; Angle-ply; Composite; Laminate; Layerwise constant shear-angle theory (LCST); First-order shear deformation theory

(FSDT); Classical lamination theory (CLT); Interlaminar shear deformation; Thick plate; Triangular element; Scaling effect

1. Introduction

Recent years have witnessed an increasing use of

advanced composite materials (e.g., graphite/epoxy,

boron/epoxy, Kevlar/epoxy, graphite/PEEK, etc.),

which are replacing metallic alloys in the fabrication of

load-bearing plate-type structures because of many

beneficial properties, such as higher strength-to-weight

ratios, longer fatigue (including sonic fatigue) life,better stealth characteristics, enhanced corrosion resis-

tance, and, most significantly, the possibility of optimal

design through the variation of stacking pattern, fiber

orientation, and so forth, known as composite tailor-

ing. The advantages that accrue from these properties

are, however, not attainable without paying for the

complexities that are introduced by various coupling

effects. Furthermore, since the matrix material is ofrelatively low shearing stiffness as compared to the fi-

bers, a reliable prediction of the response of these

laminated shells must account for interlaminar (trans-

verse) shear deformation or cross-sectional warping of

individual layers, in contrast to the Kirchhoff or

Mindlin hypothesis. The former, known as the classical

* Tel.: +1-801-581-6863; fax: +1-801-581-4816.

E-mail address: [email protected] (R.A. Chaudhuri).

0263-8223/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruct.2004.01.002

lamination theory (CLT), neglects the interlaminarshear deformation altogether, while the latter, called

the first-order shear deformation theory (FSDT) as-

sumes constant transverse shear deformation through

the entire thickness of the laminate. More recently,

various refined or higher-order shear deformation the-

ory (HSDT) based solutions have become available in

the literature. Basset [1] appears to have been the first

to suggest that the in-plane displacements can be ex-panded in power series of the thickness coordinate.

Following Basset’s lead, second- and higher-order shear

deformation theories (HSDT), assuming transverse in-

extensibility and continuous inplane displacements

through the thickness of thick laminates, have been

developed as special cases of the above, e.g., Refs. [2,3].

A detailed review of the literature and a double Fourier

series solution to a general type boundary-value prob-lem pertaining to HSDT-based laminates are available

in Chaudhuri and Kabir [4].

Noor and Burton [5] have presented extensive surveys

on shear deformation theories and computational mod-

els relating to laminated plates. Exact three-dimensional

elasticity solutions for rectangular cross-ply plates for

a specific type of simply supported boundary condi-

tion are due to Pagano [6], and Srinivas and Rao [7].Approximate thick laminate theories can be classified

mail to: [email protected]

Nomenclature

a, b length and width of a laminated plate

½Bj� strain–nodal displacement relation matrix for

the jth composite plate element½CðiÞ�, ½GðiÞ� elastic stiffness matrices of the ith aniso-

tropic lamina in inplane stretching/shear and

transverse shear, respectively

fdjg nodal displacement vector of the jth com-

posite plate element�di distance from the bottom (reference) surface

E1, E2 Young’s moduli of an orthotropic lamina in

the direction of fibers and normal to the fi-bers, respectively

Fj consistent load vector of the jth composite

plate element

G12 inplane shear modulus of an orthotropic

lamina

G13, G23 transverse shear moduli of an orthotropic

lamina

½Kj� stiffness matrix for the jth composite plateelement

N total number of layers or laminae

Nd number of subdivisions in a finite element

model

q0 applied uniformly distributed load (normal

pressure) on the top surface of a laminated

plate

qjðx1; x2Þ applied surface load on the top surface ofthe jth composite plate element

ti, t thickness of the ith lamina and laminated

plate, respectively

U strain energy of a laminated plate

U strain energy per unit area

U ðiÞB strain energy in bending and twisting of the

ith layer

U ðiÞS strain energy in transverse (interlaminar)

shear of the ith layer

ui components of the displacement vector,

i ¼ 1; 2; 3W potential due to external conservative forces

w deflection of a plate

xi cartesian coordinates, i ¼ 1; 2; 3cðiÞ12 inplane (engineering) shearing strain at a

point inside the ith layer

cðiÞ13, cðiÞ23 transverse (engineering) shearing strains at a

point inside the ith layer

Dj reference surface area of the jth element

eðiÞk‘ inplane components of the strain tensor at a

point inside the ith layer; k; ‘ ¼ 1; 2m12, m13, m23 major Poisson’s ratios of an orthotropic

lamina

P total potential energy functional

�sðNþ1Þ13 , �sðNþ1Þ

23 , �rðNþ1Þ33 , �sð1Þ13 , �s

ð1Þ23 , �r

ð1Þ33 applied distributed

forces over the top and bottom surfaces of a

N -layer plate

�rðiÞnnðx3Þ, �s

ðiÞnCðx3Þ, �s

ðiÞn3ðx3Þ applied forces at a boundary

distributed through the thickness of the ithlayer

f/g shape functions

72 R.A. Chaudhuri / Composite Structures 67 (2005) 71–84

into two categories: (a) discrete layer approach, and (b)

continuous inplane displacement through thickness, e.g.

the CLT, FSDT and HSDT, mentioned above. The

former approach, e.g., the layerwise constant shear-angle

theory (LCST) or the zig-zag theory, first introduced by

Mau et al. [8] appears to be quite suitable for numerical

methods, such as the degenerate type finite element

methods (FEM), although a number of analyses onshear-flexible laminated plates have been performed

using FSDT-based finite elements, e.g., Spilker et al. [9].

Seide [10] has utilized the LCST in the derivation of an

exact solution to an infinitely long laminated strip

problem.

The LCST-based solution due to Mau et al. [8] for

laminated thick plates has used a quadrilateral element

shape, and assumed stress hybrid finite element method(FEM). Although this method has yielded results for

certain simpler laminated plate problems, it suffers from

certain difficulties, such as the presence of spurious

kinematic modes [11]. More recently, Chaudhuri and

Seide [12], and Seide and Chaudhuri [13] have developed

a quadratic triangular element, based on the LCST, and

an assumed displacement potential energy approach, for

analyses of laminated plates and shells, respectively.

Example problems analyzed by Chaudhuri and Seide

[12] include a symmetric cross-ply [90�/0�/90�] infinitelylong strip and a three-layer symmetric cross-ply square

plate. Additionally, the special case of a homogeneousisotropic plate has been presented by Chaudhuri [14].

LCST-based results on angle-ply plates, computed using

this triangular element, are not available in the litera-

ture, which is the primary objective of the present

investigation. The specific goals of the present analysis,

which assumes quadratic shape functions, are to (i)

obtain satisfactory convergence of displacements and

moments, (ii) study the effect of triangulation pattern onconvergence, (iii) investigate the effect of numerical

integration on convergence, and (iv) study the effects of

fiber orientation angle and length-to-thickness ratio on

computed displacements and stresses (or moments).

R.A. Chaudhuri / Composite Structures 67 (2005) 71–84 73

2. Theoretical background and finite element formulation

Fig. 1 shows a N -layer laminated composite trian-

gular element with its bottom surface designated as thereference surface. The local or element coordinates are

denoted by xa, a ¼ 1; 2; 3, while the corresponding glo-

bal or plate coordinates are represented by x, y, z. Thebasic assumptions underlying the formulation of a

laminated thick plate element, based on discrete layer or

zig-zag approach are: (i) small deformation, (ii) linear

anisotropic elasticity, (iii) transverse inextensibility and

(iv) layerwise constant shear angle hypothesis. By virtueof these assumptions, the inplane strains at a point in-

side the ith lamina are given as follows [12]:

eðiÞðx3Þ� �

¼ AðiÞb ðx3Þ

h iAðiÞt ðx3Þ

h ih i �eðiÞn o�eðiþ1Þn o

8<:

9=;; ð1Þ

in which ½AðiÞb ðx3Þ� and ½AðiÞ

t ðx3Þ� are as given by Eq. (A.1)in the Appendix A, while

eðiÞðx3Þ� �T ¼ eðiÞ11ðx3Þ eðiÞ22ðx3Þ cðiÞ12ðx3Þ

� �;

for i ¼ 1; . . . ;N ; ð2aÞ

and

�eðiÞn oT

¼ �eðiÞ11 �eðiÞ22 �cðiÞ12

n o; for i ¼ 1; . . . ;N þ 1:

ð2bÞ

The interlaminar or transverse shear strains in the ithlamina given by

cðiÞ� �T ¼ cðiÞ13 cðiÞ23

n o; for i ¼ 1; . . . ;N ð3Þ

do not vary with x3 within a lamina. The transverse

shear strain components, cðiÞk3 , k ¼ 1; 2, can be written as

cðiÞk3 ¼ �u3;k þ1

tið�uðiþ1Þ

k � �uðiÞk Þ: ð4Þ

The total potential energy of the N -layer laminated

anisotropic plate can now be written as follows:

P ¼ U þ W ; ð5Þ

Fig. 1. (a) A laminated composite plate element of triangular plan-

form. (b) A layer element.

where U , the strain energy, is given by

U ¼XNi¼1

ðU ðiÞB þ U ðiÞ

S Þ; ð6Þ

in which the bending contribution to the lamina strain

energy is given as follows:

U ðiÞB ¼ 1

2

Zx2

Zx1

Zx3

eðiÞðx3Þ� �T½CðiÞ� eðiÞðx3Þ

� �dx3 dx1 dx2;

ð7aÞ

and the corresponding interlaminar or transverse shear

contribution is

U ðiÞS ¼ 1

2

Zx2

Zx1

Zx3

cðiÞðx3Þ� �T½GðiÞ� cðiÞðx3Þ

� �dx3 dx1 dx2;

ð7bÞ

where ½CðiÞ� and ½GðiÞ�, the lamina elastic stiffness matri-

ces as described by Eqs. (A.3a) and (A.3b), respectively,

are related to Young’s and shear moduli, and Poisson’s

ratios.

With the lamina strain–displacement and constitutiverelations given by Eqs. (1)–(4) and (A.3), respectively,

the strain energy for the laminated plate can be written

from Eqs. (6) and (7), on integration with respect to x3,as

U ¼Zx2

Zx1

U dx1 dx2; ð8Þ

wherein U , the strain energy of the laminated plate per

unit area of the reference (bottom) surface, is given by

U ¼ 1

2f�egT½D�f�eg; ð9Þ

in which

f�egT ¼ f�eg fcg� �

; ð10Þ

with

f�egT ¼ �eð1Þn oT

� � � �eðiÞn oT

� � � �eðNþ1Þn oT

� �;

ð11aÞ

and

fcgT ¼ cð1Þ� �T � � � fcðiÞgT � � � fcðNÞgT

n o: ð11bÞ

For a N -layer plate, ½D� is a ð5N þ 3Þ � ð5N þ 3Þ matrix

given by Eqs. (A.4)–(A.6).

If the applied surface and edge forces are conserva-

tive, the potential due to these forces is given by


W ¼ �Z Z

Sð�sðNþ1Þ

13 �uðNþ1Þ1

hþ �sðNþ1Þ

23 �uðNþ1Þ2

þ �rðNþ1Þ33 �uðNþ1Þ

3 Þ � ð�sð1Þ13 �uð1Þ1 þ �sð1Þ23 �u

ð1Þ2

þ �rð1Þ33 �u

ð1Þ3 ÞidS �

ZC0

XNi¼1

Z �diþ1

x3¼di

½�sðiÞnCðx3ÞuðiÞC ðx3Þ

þ �rðiÞnnðx3ÞuðiÞn ðx3Þ þ �sðiÞn3ðx3Þu

ðiÞ3 �dx3 dC; ð12Þ

in which

�di ¼Xi�1

m¼1

tm; �d1 ¼ 0; �dNþ1 ¼ t: ð13Þ

The quadratic shape function used for the triangular

element is best expressed in terms of area coordinates as

shown below:

f/gT ¼ f1ð2f1 � 1Þ 4f1f2 f2ð2f2 � 1Þ 4f2f3 f3ð2f3 � 1Þ 4f3f1f g; ð14Þ

where fk, k ¼ 1; 2; 3 represents area coordinates [15].

Finally, the strain energy of the jth triangular element

can be written in the form [12]:

Uj ¼1

2fdjgT½Kj�fdjg; ð15Þ

in which the corresponding element stiffness matrix is

given by

½Kj� ¼ZDj

Z½½Bj�T½Dj�½Bj��dD; ð16Þ

where ½Bj� is a matrix given by Eqs. (A.7)–(A.11), while

fdjg is a column vector, given as follows:

fdjgT ¼ ð�u1Þð1Þ1 ð�u2Þð1Þ1 � � � ð�u1Þð1Þ6 ð�u2Þð1Þ6 � � � ð�u1ÞðNÞ6 ð�u2ÞðNÞ

6 ðu3Þ1 � � � ðu3Þ6n o

: ð17Þ

Fig. 2. A rectangular laminated plate.

The subscripts outside the parentheses on the right side

of Eq. (17) represent the kth node of the element inter-

face for k ¼ 1; . . . ; 6.Similar operations as above on Eq. (12) yield the

potential due to applied conservative forces on the top

and bottom surfaces of the plate element as well as those

acting on its edge through its thickness. For example,

the potential due to distributed normal load acting onthe top surface of the jth element of the N -layer plate is

obtained by substituting

�rðNþ1Þ33 ¼ qjðx1; x2Þ; ð18aÞ

and

�rð1Þ33 ¼ �sð1Þ13 ¼ �sð1Þ23 ¼ �sðNþ1Þ

13 ¼ �sðNþ1Þ23 ¼ �rðiÞ

nnðx3Þ

¼ �sðiÞnCðx3Þ ¼ �sðiÞn3ðx3Þ ¼ 0; for i ¼ 1; . . . ;N ; ð18bÞ

into Eq. (12), yielding

Wj ¼ �fdjgTfFjg; ð19Þ

where fFjg, the consistent load vector of the jth com-

posite-element, is given by

fFjgT ¼ f0g fPgj� �

; ð20Þ

in which {0} is a 12ðN þ 1Þ null row-vector, while fPgj isa 6-element row-vector, given by

fPgj ¼ P1j � � � Pkj � � � P6jf g; ð21Þ

with

Pkj ¼1

2

ZDj

Zqjðx1; x2Þ/kðx1; x2ÞdD; for

j ¼ 1; . . . ;N ; k ¼ 1; . . . ; 6: ð22Þ

3. Numerical results and discussions

Fig. 2 shows a laminated rectangular plate. The fol-

lowing example problems, pertaining to the deformation

of square (a ¼ b ¼ 25:4 cm (10 in)) angle-ply plates

Fig. 3. Finite element mesh for a square ±45� plate.


subjected to uniform transverse load, q0, will serve to

not only illustrate the validity of the finite element

procedure presented in the preceding section, but also

present hitherto unavailable benchmark numericalsolutions. The following material (lamina) properties for

a representative high modulus fiber reinforced compos-

ite are considered:

E1

E2

¼ 40;G12

E2

¼ G13

E2

¼ G23

E2

¼ 0:5; m12 ¼ m13 ¼ m23 ¼ 0:25;

where E1 and E2 represent the inplane Young’s moduli

in the x1 and x2 directions, respectively, while m12 is the

inplane major Poisson’s ratio, and m13 and m23 denote thesame in the x1–x3 and x2–x3 planes, respectively. G12

denotes the inplane shear modulus, while G13 and G23

represent the transverse shear moduli in the x1–x3 and

x2–x3 planes, respectively. Before proceeding to compute

the numerical results for angle-ply laminates, the

numerical results for cross-ply plates [12] have been

reproduced first, which lends some credence to the

accuracy of what follows. The following normalized

quantities are defined:

�w ¼ 103E2ðt=NÞ3

�q0a4w; ð23aÞ

�u ¼ 10E2ðt=NÞ2

�q0a3u; ð23bÞ

ð�rðiÞx ; �rðiÞ

y �sðiÞxy Þ ¼

10ðt=NÞ2

�q0a2ðrðiÞ

x ; rðiÞy sðiÞxy Þ; ð23cÞ

ðMx;MyÞ ¼10

�q0a2ðMx;MyÞ; ð23dÞ

in which �q0 ¼ q0, for a two-layer balanced antisymmet-

ric angle-ply plate, while �q0 ¼ q0=2, for a three-layerunbalanced symmetric angle-ply laminate numerically

investigated below.

3.1. Example 1: Two-layer antisymmetric angle-ply plates

under uniform pressure

As an important check on the accuracy of the trian-

gular element under consideration, the problem of a

uniformly loaded two-layer antisymmetric balanced

angle-ply ð��hÞ plate is investigated in detail. The layers

are of equal thickness, t=2. The plate is simply supported

on all four edges so that transverse displacements there

vanish. Normal and parallel edge inplane displacements

at the top and bottom surfaces of the plate are free tooccur. Whitney [16] has solved the same problem using

the classical lamination theory (CLT) in which the effect

of transverse shear deformation is neglected. Spilker

et al. [9] have obtained solutions using the assumed

stress hybrid finite element method based on the first

order shear deformation theory (FSDT), and have

achieved, by modeling with quadrilateral elements,

reasonably close agreement with a corrected version of

Whitney’s [16] results. However, as will also be seen

below, their triangular elements do not yield results of

desired accuracy.Because of apparent lack of symmetry conditions due

to the coupling of bending and inplane shear (or

equivalently, twisting–extension coupling), the entire

plate has to be modeled except for h ¼ �45�, in which

case only half the plate is modeled (Fig. 3).

3.2. Triangulation pattern

Before investigating the convergence of the present

triangular element, it is important to determine which

geometrical configuration or triangulation pattern yields

superior performance with the same number of degreesof freedom. Two triangulation patterns are possible: (a)

unidirectional diagonals (Fig. 4a) and (b) bi-directional

diagonals (Fig. 4b). Strang and Fix [17] have invoked

the following theorem, which states that: If Sh is of de-

gree k0 � 1 on a regular mesh, there exist non-negative

definite matrices K, such that for every u in Hk0

h2ðs0�k0Þmin

Sh

ju�uhj2s0 !

Xja0 j¼jb0 j¼k0

Ka0b0

S0

ZðDa0uÞ

ZðDb0uÞdx;

ð24Þ

and have shown for the two-dimensional case that the

pattern (a) is superior to the pattern (b) by a factor of

1.25. This is tested here via a numerical experiment for

the two-layer ±45� angle-ply laminate using a 2 · 2 mesh

and quintic order of quadrature. The numerical experi-

ment favors the pattern (a) over (b) by a factor of 1.68.

Since both the theory and numerical experiment favorsthe pattern with unidirectional diagonals, the results

presented here are computed using this pattern only.

Fig. 4. Triangulation pattern with (a) unidirectional and (b) bidirec-

tional diagonals.

Fig. 5. (a) Effect of numerical integration on the convergence of nor-

malized central deflection of a square ±45� plate. (b) Comparison of

convergence of normalized central deflection computed by various fi-

nite element methods for a square ±45� plate.


3.3. Convergence

A LCST based finite element such as the present tri-

angular one is generally expected to encounter shear

locking in the thin plate regime, e.g., a=t ¼ 50. Since the

LCST based theory is primarily valid for thicker plates,

the performance of the element in the thick plate regime isgenerally much superior in the thick plate regime

(a=t ¼ 4–20). Figs. 5–13 show convergence of displace-

ments and moment resultants in the thin laminate

ða=t ¼ 50Þ. The items of interest are: (a) transverse dis-

placement or deflection, w, at the center of the laminate;

(b) the inplane x-direction displacement uð1Þðt1Þ ¼ uð2Þð0Þat x ¼ a=2, y ¼ 0; (c) the inplane y-direction displacement

vð1Þðt1Þ ¼ vð2Þð0Þ at x ¼ 0, y ¼ a=2; and ðd; eÞ the momentresultants, Mx and My , at the center of the laminate.

It should be noted that in the present theory, distri-

bution of stresses through thickness is more meaningful

than computation of moments. However, since the re-

sults of the present investigation are being compared

with those of the classical lamination theory (CLT) [16]

and first order shear deformation theory (FSDT) [9], the

moment resultant, Mx, for the two-layer angle-ply plateis computed here as follows:

Mx ¼ rð1Þx ð0Þ t

21

3þ rð1Þ

x ðt1Þt216� rð2Þ

x ð0Þ t22

6� rð2Þ

x ðt2Þt223; ð25Þ

My and Mxy can similarly be obtained by replacing rx by

ry and sxy , respectively, in Eq. (25).

The effect of numerical integration on convergence is

investigated first for �h ¼ �45� plate (Figs. 5–7). Theresults, shown in Figs. 5(a), 6(a), and 7(a), confirm that

the quadratic order (referred to as reduced integration),

which corresponds to the minimal order for bending and

reduced integration for shear terms, accelerates the

convergence for displacements and stresses. However, as

can be observed from Figs. 6(a) and 7(a), the inplane

displacements and moment resultants do not converge

monotonically if reduced integration is used. This isbecause the error due to reduced integration is not of the

same order as that due to the finite element approxi-

mation, and therefore, monotonicity of convergence

cannot be guaranteed. The quintic order (also referred

to as the full integration) ensures monotonicity of con-

vergence.

Figs. 5(b), 6(b) and 7(b) also compare the rate of

convergence and accuracy of the present triangular ele-ment based on the layerwise constant shear-angle theory

(LCST) or thick plate theory with those of three hybrid

stress elements due to Spilker et al. [9], based on the first

order shear deformation theory (FSDT) or moderately

thick plate theory. The triangular element of Ref. [9]

Fig. 6. (a) Effect of numerical integration on the convergence of nor-

malized interfacial inplane displacements of at the edge of a square

±45� plate. (b) Comparison of convergence of normalized interfacial

inplane displacements computed by various finite element methods for

a square ±45� plate.

Fig. 8. Convergence of central deflection of a square ±15� plate.

Fig. 7. (a) Effect of numerical integration on the convergence of mo-

ments of a square ±45� plate. (b) Comparison of convergence of mo-

ments computed by various finite element methods for a square ±45�plate.

Fig. 9. Convergence of interfacial inplane displacement, uint, at the

edge of a square ±15� plate.


formulated without the reduction of stress parameters,

designated hybrid triangular 1 in Figs. 5(b), 6(b) and

7(b), is shown to be too stiff to be useful in practical

applications. The same element of Ref. [9] with reduc-

tion of stress parameters, designated hybrid triangular 2in Figs. 5(b), 6(b) and 7(b), shows better convergence

performance than the one without, but the monotonicity

of convergence of the central deflection, wc and moment,

Mx;c is now lost. The quadrilateral element of Spilker et

al. [9] yields monotonic convergence of displacements

and moment resultants to the corresponding corrected

version of the CLT solutions due to Whitney [16]. The

present triangular element yields monotonic conver-gence of displacements and moment resultants, when the

quintic order of numerical integration is used. However,

for the ±45� laminate under consideration, Figs. 5–7

show overshooting of computed displacements and

moment resultants over their CLT counterparts due to

Whitney [16] and FSDT-based hybrid element results of

Spilker et al. [9].

The reason behind this overshoot over the CLT isobvious––the CLT simply ignores the shear deformation

altogether. However, the overshoot over the FSDT re-

sults can be attributed to the fact that for an angle-ply

laminate, such as the present one, not only the high

Fig. 11. Convergence of central moment, Mx;c, of a square ±15� plate.

Fig. 10. Convergence of interfacial inplane displacement, vint, at theedge of a square ±15� plate.

Fig. 12. Convergence of central moment, My;c, of a square ±15� plate.

Fig. 13. Variation of the computed central deflection with respect to

fiber orientation angle for square two-layer antisymmetric angle-ply

plates.

Fig. 14. Variation of the computed edge interfacial inplane displace-

ment, uint, with respect to fiber orientation angle for square two-layer

antisymmetric angle-ply plates.


length-to-thickness ratio, a=t, but also fiber orientation

angle in a layer, �hi, plays a significant role in interla-

minar shear deformation. The FSDT based hybrid ele-

ments due to Spilker et al. [9] fail to detect this role. To

ensure that the overshoot, mentioned earlier in the case

of a ±45� laminate, is actually due to interlaminar shear

deformation caused by the large difference in fiber ori-entation angles in two consecutive layers, (i.e.,�h2 � �h1 ¼ 90�), another angle-ply laminate, with�h ¼ �15� (i.e., �h2 � �h1 ¼ 30�), is also investigated for

convergence (see Figs. 8–12). These figures show that the

central deflection, wc, and central moments Mx;c and My;c

converge monotonically to their CLT counterparts. The

interfacial inplane displacement, vint at x ¼ 0 and

y ¼ a=2 slightly overshoots the corresponding CLT re-

sult, while uint at x ¼ a=2 and y ¼ 0 has an undershoot.

These results confirm the afore-mentioned assertion thatfor �h ¼ �45�, interlaminar shear deformation is present

even when the plate is thin. This interlaminar shear

deformation cannot be detected by the FSDT used by

Spilker et al. [9].

3.4. Effect of fiber orientation

To assess the effects of the fiber orientation angle, �h,on the finite element solution, a series of results are

obtained for various fiber orientation angles, such as�h ¼ 5�, 7.5�, 15�, 22.5�, 25�, 30�, 35�, 37.5� and 45�. Theresults for �ð90�� hÞ can be obtained from ��h angle-

ply laminate results by rotating x–y axes by 90�. Figs.13–17 show variation of displacements and momentresultants with respect to �h. These results confirm the

assertion made earlier that for j�hiþ1 � �hij � 0� or 180�(i.e., close to unidirectional anisotropic plates), the re-

sults for thin angle-ply laminates ða=t ¼ 50Þ obtained by

using the layerwise constant shear angle theory (LCST)

or thick plate theory, the first order shear deformation

Fig. 15. Variation of the computed edge interfacial inplane displace-

ment, vint, with respect to fiber orientation angle for square two-layer

antisymmetric angle-ply plates.

Fig. 16. Variation of the computed central moment, Mx;c, with respect

to fiber orientation angle for square two-layer antisymmetric angle-ply

plates.

Fig. 17. Variation of the computed central moment, My;c, with respect

to fiber orientation angle for square two-layer antisymmetric angle-ply

plates.

Fig. 18. Comparison of wðxÞ variation with x computed using the

present element with its CLT counterpart.

Fig. 19. Comparison of rxðxÞ variation with x computed using the

present element with its CLT counterpart.


theory (FSDT) or moderately thick plate theory, and theclassical lamination theory (CLT) or the thin plate the-

ory exhibit close agreements. With departure from the

unidirectional or close to it, the marked difference begins

to show up among the results of the present theory and

the remaining two, even though a=t ¼ 50 qualifies the

laminate under consideration to be a thin plate. It is

noteworthy that the central deflection, wc, and central

moments Mx;c and My;c, computed by the LCST-based

FEM are consistently higher than their FSDT and CLT-

counterparts, and the difference increases monotonically

as �h increases from 0� to 45� (see Figs. 13, 16, 17). The

interfacial inplane displacement, vint at x ¼ 0 andy ¼ a=2, computed by the present theory is also higher

than its FSDT and CLT counterparts (Fig. 15). In

contrast, the interfacial inplane displacement, uint at

x ¼ 0, y ¼ a=2, given by the present theory is lower than

the corresponding uint values computed using the other

two theories, except for a small range of �h around 45�(Fig. 14). These results confirm that unlike the FSDT,

the LCST can detect the cross-sectional warping causedby drastic change of elastic constants at the interface of

layers of even a thin ða=t ¼ 50Þ antisymmetric angle-ply

laminate.

3.5. Displacement and stress distribution

Figs. 18 and 19 show variation, along the central line

y ¼ a=2, of displacements and stresses computed using

12 · 12 mesh of the present triangular element with

quintic order of numerical integration. Figs. 18 and 19

compare w and r1xð0Þ, respectively, along y ¼ a=2 com-

puted by the present finite element method with those,calculated using 100 terms of the Fourier series solutions

due to Whitney’s [16] CLT solution. Figs. 18 and 19

Fig. 20. rx;c distribution through the thickness of a square ±45� plate.Fig. 21. Variation of the computed normalized central deflection with

respect to aspect ratio, a=t for square ±45� plates.

Fig. 22. Variation of the maximum normalized inplane interface dis-

placement with respect to aspect ratio, a=t for square ±45� plates.

Fig. 23. Variation of the normalized inplane stress computed at the

center with respect to aspect ratio, a/t for square ±45� plates.


show that w and rx are symmetric with respect to the

plate centerline.

Fig. 20 shows the variation of the normalized stress,�rðiÞx ðzÞ in a ±45� laminate through its thickness, and

compares the results computed using the present LCST-based FEM with those obtained using 100 terms of the

Fourier series solution due to Whitney’s [16] CLT

solution. The slight difference between the two sets of

results, presented in Figs. 18–20, can be attributed to the

presence of interlaminar shear deformation in a ±45�angle-ply laminate, which is neglected in the CLT. Fig.

20 further shows that rxðzÞ is antisymmetric about the

layer interface, where it experiences a jump.

3.6. Scaling effect

To assess the effect of length-to-thickness ratio oncomputed displacements and stresses in an angle-ply

plate, only the lamina thickness is varied, while the rest

of the geometric and material parameters remain unal-

tered. Here for a ±45� plate, a series of results are ob-

tained with layer thickness tið¼ t=2Þ ¼ 0:1, 0.15, 0.2,

0.25, 0.35, 0.5, 0.75, 1 in. The quantities of interest are

normalized central deflection, wc, given by

wc ¼103E2t3wc

�q0a4¼ N 3�wc; ð26aÞ

normalized interfacial displacement, uint at x ¼ a=2,y ¼ 0, given by

uint ¼103E2ðtÞ2

�q0a3uint ¼ 102N 2�uint; ð26bÞ

and normalized stress, given by

rð1Þx;c ð0Þ ¼

102t2rð1Þx;c ð0Þ

�q0a2¼ 10N 2�rð1Þ

x;c ð0Þ: ð26cÞ

These normalized quantities, given by Eqs. (26a–c) are

presented in Figs. 21–23, respectively. It is interesting to

observe from Figs. 21 and 23 that while for a thick

ða=t ¼ 5Þ ±45� laminate, the normalized central deflec-

tion, wc, computed by the CLT is in serious error,

error¼ 100(LCST)CLT)/LCST% being about 60%, the

corresponding error for the central stress at the bottom

surface, rð1Þx;c ð0Þ, computed by the CLT is significantly

lower, 100(LCST)CLT)/LCST% being 24.5% (ap-

prox.). Both of these normalized quantities, computed

by the LCST, decrease monotonically with the aspect

ratio, a=t. This notwithstanding, the differences betweenthe results computed by the LCST and CLT do not

completely vanish even for a bona fide thin plate ða=t ¼50Þ because of the existence of interlaminar shear

deformation explained earlier. In contrast, the normal-

ized interfacial inplane displacement, uint, computed by

the LCST at the edge point, x ¼ a=2, y ¼ 0, is lower than

its CLT-based counterpart for a very thick laminate

ða=t ¼ 5Þ, then increases with a=t, becomes larger than

its CLT counterpart, and reaches its maximum at a=t ¼20 (approx.), beyond which it slowly decreases. As be-fore, the difference between the results computed by the

LCST and CLT does not completely vanish even for a

bona fide thin plate ða=t ¼ 50Þ because of the existence

of interlaminar shear deformation.

Fig. 25. Comparison of the variation of the computed normalized

central deflection with respect to fiber orientation angle for square � �hsymmetric angle-ply plates.

Fig. 26. Comparison of the variation of the computed edge interfacial

inplane displacement, uint, with respect to fiber orientation angle for

square � �h symmetric angle-ply plates.


3.7. Example 2: Three-layer unbalanced symmetric angle-

ply plates under uniform pressure

Since convergence and accuracy of the present LCST-based quadratic triangular element have now been

established in the previous example, this section will be

primarily devoted to obtaining certain useful results.

Geometrical configuration, elastic properties, and

thickness of each layer are the same as those of the

balanced antisymmetric angle-ply plate. The only dif-

ference is that the present laminate is comprised of three

layers, stacked symmetrically with respect to the middlesurface, and the fibers are oriented at � �h, �h being po-

sitive, with respect to x-axis. Fig. 24 shows the conver-

gence of CLT-based normalized central deflection of a

thin laminate ða=t ¼ 33:333Þ computed using the

boundary continuous displacement double Fourier ser-

ies solution [18,19] with the number of terms.

In regards to the LCST-based finite element solution,

symmetric lamination permits modeling only the bottomlayer and half of the middle layer, subjected to a uni-

form pressure of �q0 ¼ q0=2. Inplane displacements van-

ish on the middle surface. All four edges are simply

supported. The entire plate planform has been modeled

here to investigate the effect of the fiber orientation

angle, �h, on central deflection, wc, interfacial inplane

displacements, uint at x ¼ �a=2, y ¼ 0, and vint at x ¼ 0,

y ¼ �a=2. These are shown in Figs. 25–27. It is inter-esting to compare these results with their balanced

antisymmetric angle-ply counterparts shown in Figs. 13–

15. The central deflection, wc, here, like the its coun-

terpart in example 1, first increases with �h, and then

decreases until �h ¼ 45� is reached. Variation of uint andvint with �h differs from the corresponding variation of the

previous example. Figs. 25–27 further show that differ-

ence between the displacements computed by the LCSTand CLT increase with the fiber orientation angle, �h, inthe range of �h considered. This can be attributed to the

increase of interlaminar shear deformation or cross-

Fig. 24. Convergence of normalized central deflection computed using

double Fourier series solution for square � �h symmetric angle-ply

plates.

Fig. 27. Comparison of the variation of the computed edge interfacial

inplane displacement, vint, with respect to fiber orientation angle for

square � �h angle-ply plates.

Fig. 28. rx;c distribution through the thickness of a �45� angle-ply

plate.


sectional warping with more drastic changes of materialproperties across a layer interface as �h increases from 0�to 45�. Fig. 28 shows variation of rxðzÞ, computed by the

LCST and CLT, through the plate thickness at the

center of the �45� plate. Other results are similar to

their balanced antisymmetric angle-ply counterparts,

and hence are not displayed here in the interest of

brevity of presentation.

4. Summary and conclusions

A recently developed C0-type triangular compositeplate element, based on the assumption of transverse

inextensibility and layerwise constant shear-angle theory

(LCST), is utilized to analyze antisymmetric and sym-

metric angle-ply plates subjected to distributed trans-

verse loading. Effect of numerical integration on the rate

of convergence of displacements and moments is inves-

tigated in detail. Numerical results demonstrate that no

shear locking has been encountered in the thin plateregime, even when the full integration scheme is em-

ployed. These results further demonstrate that use of the

reduced integration scheme, in general, accelerates the

rate of convergence of both displacements and stresses.

However, monotonicity of convergence may sometimes

be lost, when the reduced integration is used.

Comparison of the numerical results computed using

the present triangular element with their analyticalcounterparts based on the classical lamination theory

(CLT) in the thin plate regime also forms a major part

of the present investigation. Limited comparisons with

the first-order shear deformation theory (FSDT) results,

computed using assumed stress hybrid finite elements,

are also presented. Numerical results presented also in-

clude the effect of fiber orientation angle on the dis-

placements and moments for thin laminates. The LCST-

based results for an antisymmetric angle-ply ð�h ¼ �45�Þplate show that the interlaminar shear deformation is

present even when the plate is reasonably thin

ða=t ¼ 50Þ. This interlaminar shear deformation cannot

be detected by the FSDT. The computed results confirm

that unlike the FSDT, the LCST can detect the cross-

sectional warping caused by drastic change of elastic

constants at the interface of layers of even thin

ða=t ¼ 50Þ angle-ply laminates made of advanced com-posite materials, such as graphite–epoxy and boron–

epoxy.

Finally, effect of thickness on the computed trans-

verse displacement (deflection), interfacial inplane dis-

placement and inplane stress is also thoroughly

investigated. The computed results for antisymmetric

angle-ply ð�h ¼ �45�Þ plates show that while the trans-

verse displacement of a thick laminate ða=t ¼ 5Þ com-puted by the CLT is in serious error (�60%) when

measured against the LCST, the corresponding error in

the computed stress is much lower (<25%). These hith-

erto unavailable results are expected to serve as bench-

mark numerical solutions for future comparisons with

more refined thick laminate theories.

Appendix A

For a N -layer laminated plate, ½AðiÞb ðx3Þ� referred to in

Eq. (1) is given as follows:

AðiÞb ðx3Þ

h i¼

aðiÞb ðx3ÞaðiÞb ðx3Þ

aðiÞb ðx3Þ

2664

3775;

for i ¼ 1; . . . ;N ; ðA:1Þ

in which

aðiÞb ðx3Þ ¼ ½Hðx3 � �diÞ � Hðx3 � �diþ1Þ� 1

� x3 � �di

ti

!;

ðA:2Þ

½AðiÞt ðx3Þ� can be obtained by replacing the subscript b

by t.The inplane stress–strain relation at a point inside the

ith layer is given by

rðiÞðx3Þ� �

¼ ½CðiÞ� eðiÞðx3Þ� �

; for i ¼ 1; . . . ;N ; ðA:3aÞ

while the transverse or interlaminar shear stress–strain

relation for the ith layer can be written as follows:

fsðiÞðx3Þg ¼ ½GðiÞ� cðiÞðx3Þ� �

; for i ¼ 1; . . . ;N ; ðA:3bÞ

½D�-matrix referred to in Eq. (9) is given by


½D� ¼ ½DB� ½0�½0� ½DS�

� �; ðA:4Þ

in which

½DB� ¼

Dð1Þb

h iDð1Þ

t

h iDð1Þ

bt

h iDð1Þ

t

h iþ Dð2Þ

b

h iDð2Þ

bt

h iDði�1Þ

bt

h iDði�1Þ

t

h iþ ½DðiÞ

b � DðiÞbt

h iDðN�1Þ

bt

h iDðN�1Þ

t

h iþ DðNÞ

b

h iDðNÞ

bt

h iDðNÞ

bt

h iDðNÞ

t

h i

26666666664

37777777775;

ðA:5aÞ

and

½DS� ¼½Dð1Þ

s �½DðiÞ

s �½DðNÞ

s �

24

35; ðA:5bÞ

while the sub-matrices are defined as shown below:

½DðiÞb � ¼ ½DðiÞ

t � ¼ 2½DðiÞbt � ¼

ti3½CðiÞ�;

for i ¼ 1; . . . ;N ; ðA:6aÞ

½DðiÞs � ¼ ti½GðiÞ�; for i ¼ 1; . . . ;N ; ðA:6bÞ

½Bj�-matrix referred to in Eq. (16) is given by

½Bj� ¼

½R� ½0�½R� ½0�

½R� ½0�½M ð1Þ� ½N ð1Þ� ½T �

½M ðiÞ� ½N ðiÞ� ½T �½M ðNÞ� ½N ðNÞ� ½T �

26666664

37777775; ðA:7Þ

whose submatrices are given as follows:

½R� ¼ ½½R1� � � � ½Rk� � � � ½R6��; k ¼ 1; . . . 6; ðA:8aÞwith

½Rk� ¼/k;1 0

0 /k;2

/k;2 /k;1

24

35; ðA:8bÞ

½T � ¼ ½½T1� � � � ½Tk� � � � ½T6��; k ¼ 1; . . . ; 6; ðA:9aÞ

½Tk� ¼/k;1

/k;2

� �: ðA:9bÞ

½N ðiÞ� ¼ ½½N ðiÞ1 � � � � ½N ðiÞ

k � � � � ½N ðiÞ6 �; k ¼ 1; . . . ; 6;

i ¼ 1; . . . ;N ; ðA:10aÞ

with

½N ðiÞk � ¼

/kti

/kti

" #; ðA:10bÞ

while

½M ðiÞ� ¼ �½N ðiÞ�; i ¼ 1; . . . ;N : ðA:11Þ

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Analysis of laminated shear-flexible angle-ply plates

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