4:4,"? ’

01

. ‘

g‘
. .1 .
.l~-t{(‘\—~
:>A ~41

'(

\‘
,4,"
Pa

5. t
".r
»

k‘
iii
A M
m

I. ,
‘l
y~n¢
-}‘

mam-r

 

.. ' c K.”
«no? ix-‘l' .7

1.! 41...“. 4:,u?

‘.’-'¥;':':hr- s

i.- rtéfia‘s ¢ 1'.

,5 't: ‘

 

“1’i;§1

» .‘ 5x
‘ * .» .392»
a ‘

(-g‘vé c

\

5‘ 15.: "-
“Mug:- 31‘
. {- ?‘ .§,L ' "

"l - ,v- ‘0‘-

~L ~ . . *235-3.‘
4%! . “ .. . .u ”3}
'1“; ., . - . -- ~ - . " p " ~~ 2w;
'aLXAmFA... '~ _ . , ~ . ‘ , Pr 9""
K 4.57;" 53". v - ‘ ~ > w... {a 3..
.42.,3- , V 3.“! ‘3?

'Jr m‘v‘

.vi‘a‘fi.

1.. .3

.1, .
“2.3%

K
a.
Mam

4'9

‘1‘.
\

J 33‘

o go:
76;:

.i >
n».

c

“aw «-
1' M' "

.1?
. \é‘

,
l4“

a: :55"
-.- ‘ -31 .
afiaéfi
' ’fi'hw
r ‘4:

,.A.

“I" ”‘2'
. .- g -

r v _- 4..

1” y I" r _ J l 7 5... .. -. ' [10,333.33

rm. n ‘ f “.1.-

u at l k

J32"

‘ fi- mar,

,. . .. .,.

,y . -. \..

3". Hum.” ‘
nqhvhmm .
m“? " "my ,‘ ,
‘ _ ,__, - , ‘ .-....
- . . asap.- -.er w-
m t, 3'. an?)

a ”x

,
mi... .t .n .
. ‘1“ 5"») J‘My h" n‘
- -........-a"..yn on...
nun—:4 I w
r rip»!
it 7" N
‘ . ."‘. “.3.-
‘ 1.5L." 3.
tvy-w-m n. .V”.x
.ormgr .u... .a .
u, a. he»
. ‘-

‘16

.. . 7;, ."‘:t'

.V ‘ . .. 2“: .3"-

;. (fl {‘1' J ,. ”hf/urn.” J. J, W . .: ugly}

. .x .‘ ‘ ‘ Jib-"y" “r I i " I
., m.” 1. M ."x

 

 

“ 1:

Eu bifl‘g-ol In

,‘vfii

 

CHIGANS

l JIIIUIHIUIIHIIIIHIJWIIIIHMWIHIIW

23 00892 9774

INN”!

 

 

 

This is to certify that the

dissertation entitled

A Study of the Interlaminar Stress Continuity Theories

for Composite Laminates

presented by

Chun-Ying Lee

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in Mechanics

 

 

Q I
Major professor E:

 

Date May /7, /qql

MS U is an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

M

r 'w—\
LIDRARY
Michigan State

1 University

'L__ A

v—_

 

 

PLACE IN BET URN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.

DATE DUE DATE DUE DATE DUE

      
 

 

 

 

 

 

 

 

 

 

h
l

 

 

 

 

 

 

 

 

 

 

 

 

 

l

MSU Is An Affirmative Action/Equal Opportunity Institution
. cidrcMma-oi

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-7 m __‘_..—. ._

 

A STUDY OF THE WAR STRESS CON'I'INUITY 'I'HBORIES
FOR COMPOSITE LAMINATES -
By

Chun-Ying Lee

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR 0F PHIIDSOPHY
in
Engineering Mechanics
Department of Metallurgy, Mechanics and Materials Science

1991

ABSTRACT

A STUDY OF THE INTERLAMINAR STRESS CONTINUITY THEORIES
FOR COMPOSITE LAMINATES

By
Chun-Ying Lee

In this study, two stress continuity theories are presented. The first one, named in-
terlaminar sness continuity theory (ISCT), accounts for the variation of transverse dis-
placement through the laminate thickness. The continuity of interlaminar shear stresses
and normal stress across the laminate interfaces and traction conditions on laminate sur-
faces are satisfied exactly. The second. interlaminar shear stress continuity (ISSCT), sim-
plifies ISCT by assuming constant transverse displacement through the thickness. Thus,
only the continuity of interlaminar shear stresses and shear traction conditions on laminate
surfaces are enforced. The merit of these stress continuity theories is the direct calculation
of interlaminar Stresses from constitutive equations instead of equilibrium equations. The
numerical examples for composite laminates with aspect ratio higher than five in cylindri-
cal bending and bidirectional bending using both theories show excellent accuracy com-
pared with elasticity solutions. ISCT provides significant improvement over ISSCT for
composite analysis only when the aspect ratio is lower than five. The comparison among
other displacement-based laminate theories and present theories is also performed.

Techniques to reduce the computational efi‘ort for these stress continuity theories

are proposed in response to the composite analysis of many-layer laminate. The layer re-

duction technique provides a methodology to retain good acctnacy while reduces the num-
ber of degree-of-freedom in composite analysis using present theories.

The further applications of ISSCT in composite analysis, e.g., vibration, buckling,
nonlinear bending, nonlinear vibration. and me—edge stresses are Studied. The associated
numerical examples Show the feasibility and potential of using this new theory in the
study of composite laminates.

To Mei-wen

iv

ACKNOWLEDGMENTS

First, I wouldliketoexpreesmydeep gratitudeandappreciation tomy majo'radvi-
sor, Dr. Dahsin Liu, for his true friendship, constant encotn'agement, and guidance.

I am also indebted to Dr. Nicholas J. Altiero for his moral support and endeavour
to do the best for students. -

Thanks also extend to my college friends at MSU for their hearty encouragement
during the course of this study. In addition, the susrained discussion and suggestion by Mr.
Xianqiang Lu is acknowledged.

Finally, but not the least, I thank the moral support and encouragement of my fam-
ily that made the difficult time easier to overcome. Especially, a great debt is owed to my

wife, Mei-wen, for her undersranding and patience during these five years.

TABLES OF CONTENTS

List of Tables

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

List of Figures x
Chapterl - Introduction- 1
1.1 Motivation ‘1

1.2 Literature Review 3

1:3 Present Work 6

- Chapter 2 - Interlaminar Stress Continuity Theories . - 8
2.1 Introduction ' 8
2.2 Interlaminar Stress Continuity Theory (ISCT) - 8

2.3 Inter-laminar Shear Stress Continuity Theory (ISSCT) - - 17
2.4 Closed-Form Solution ' 21
2.5 Finite Element Solution 23
Chapter 3 - Assessments of the Stress Continuity Theories 24
3.1 Introduction 24
3.2 Numerical Examples for Stress Continuity Theories 24
3.2.1 Laminates under Cylindrical Bending 24

3.2.2 Laminates under Bidirectional Bending 33

3.3 Comparison of Difi'erent Laminate Theories 49
3.3.1 Number of Degree-of-Freedom 49

3.3.2 Recovery of Transverse Stresses 52

3.3.3 Closed-Form Solutions for Difi‘erent Laminate Theories - 53

3.3.4 Summary 59

 

 

Chapter 4 - Techniques for Layer Reduction - - - ............................... 67

 

 

 

 

 

 

4.1 Introduction ......... - - u - - - 67
4.2 Fundamental Techniques - - - -- - _ 67
4.3 Numerical Examples - 66
4.3.1 Cylindrical Bending ...... - - 71
4.3.2 Bidirectional Bending -- - 71
4.4 Discussions -- - . 75

Chapter 5 - Applications of ISSCT in Vibration, Buckling, and Nonlinear Analysis 84

 

 

 

 

 

 

 

 

 

 

 

 

 

5.1 Introduction -- - - - - - ..... - 84
5.2 Natural Vibration -- - - -- ......... 84
5.3 Critical Buckling Load - -- - - 87
5.4 Nonlinear Bending ............................................................................................ 96
5.4.1 Formulation of Nonlinear Equation - -- - _- ....... 96

5.4.2 Laminates Subjected to Transverse Loadings - ............... 100

5.4.3 Laminates Subjecred to Inplane Loadings . - ...... 106

5.5 Large-Amplitude Vibration -- -- - - - - - 109
5.6 Free-Edge Snesses . - - - ...... - ........... 114
Chapter 6 - Conclusions and Recommendations . ...... - 119
6.1 Conclusions -- -- - 119
6.2 Recommendations ............. -- .............. 120
Appendices _ - - -_ -- - - - - 122
Appendix A o [E] and {q} Manices - - 122
Appendix B - The Equivalence of ISSCT and HSDT for One-Layer Laminate ....... 126
List of References ........................................................................................................... 128

LIST OF TABLES

Table 3.1 - Results of a simply-supported [0/90] laminate under cylindrical bending by us-
ing ISSCT 27

Table 3.2- Results of a simply-supported [0190/0] laminate under cylindrical bending by
using ISSCT .......... 28

Table 3. 3- Results of a simply-supported [0/90] laminate under cylindrical bending by us-
ing ISCT 30

Table 3.4- Results of a simply-supported [0/90/0] laminate under cylindrical bending by
using ISCT 31

Table 3.5 - Closed-form solutions of a simply-supported [0/90/90/0] square (a=b) lami-
nate under bidirectional bending by using ISSCT - 44

Table 3.6 - Closed-form solutions of a simply-supported [0/90/0] rectangular (3a=b) lam-
inate under bidirectional bending by using ISSCT - - - _- 45

Table 3.7 - Closed-form solutions of a simply-supported [0/90/90/0] square (a=b) lami-
nate under bidirectional bending by using ISCT 46

Table 3.8 - Closed-form solutions of a simply-supported [0/90/0] rectangular (3a=b) lam-
inate under bidirectional bending by using ISCT 47

Table 3.9 - Finite element solutions of a simply-supported [0/90/90/0] square (a=b) lami-
nate under bidirecrional bending by using ISSCT - - 48

Table 3.10 - Comparison of difi'erent laminate theories for an n-layer laminate ............. 51

Table 3.11 - Closed-form solutions of a [0/90] laminate under cylindrical bending by us-
ing difi'erent laminate theories 54
Table 3.12 - Closed-form solutions of a [0/90/90/0] laminate under cylindrical bending by
using difierent laminate theories - 56
Table 3.13 - Closed-form solutions of a square (a=b) [0/90/90l0] laminate under bidirec-
tional bending by using different laminate theories ...... 60

Table 3.14 - Closed-form solutions of a rectangular (3a=b) [0/90/0] laminate under bidi-
rectional bending by using different laminate theories _____ _ -_ 61

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 5.1 - Normalized fundamental frequency 1., of a simply-supported [0/90/90/0]

 

square laminate 89
Table 52 - Normalized first buckling load 1., of a simply-supported [0/90/90/01 square
laminate 93

 

Table 5.3 - Normalized first buckling load it, of a simply-supported [0190] square laminate
- 94

 

Table 5.4 - Normlized first buckling load it. of a simply-supported [0/90l0l90/0/90]
square laminate 95

 

LIST OF FIGURES

 

 

 

 

 

 

 

 

Figln'e 2.1 - Coordinate system and displacement variables - 10
Figure 3.1 - Simply-supported composite laminate under cylindrical bending .............. 25
Figlne 3. 2'- Normalized transverse displacements at midspan for [0], [0190], and [0/90/0]
laminates with difierent aspect ratios ..... _ ..32
Figure 3.3 - Normalized inplane displacement i (0) of a simply-supported [0/90] laminate
with S=4 under cylindrical bending - - 34
Figure 3.4 - Normalized inplane normal stress 6‘ (1/2) of a simply-supported [0/90] lami-
nate with S=4 under cylindrical bending - 35
Figure 3.5 - Normalind transverse shear Stress in (0) of a simply-supported [0/90] lami-
nate with S=4 under cylindrical bending 36
Figure 3.6 - Normalized n'ansverse normal Stress 6, (1/2) of a simply-supported [0/90]
laminate with S=4 under cylindrical bending 37
Figure 3.7 - Normalized inplane displacement 5(0) of a simply-supported [0/90/0] lami-
nate with S=4 under cylindrical bending - 38
Figure 3.8 - Normalized inplane normal stress 6‘ (1/2) of a simply-supported [0/90/0] lam-
inate with S=4 under cylindrical bending 39

Figure 3.9 - Normlized transverse shear stress 3,,(0) of a simply-supported [0190/0]
laminate with S=4 under cylindrical bending 40

Figure 3.10 Normalized transverse normal stress 6 (1/2) of a simply-supported [0/90/0]
laminate with S=4 under cylindrical bending

Figure 3.11- Simply-supported composite lanrinate under bidirectionally sinusoidal load-
ing - - 42

 

 

 

Figure 3.12 - Normalized n'ansverse shear stresses at the edge of a simply-supported [0/
90] laminate with S=4 by using difi'erent laminate theories. (a) constitutive
equations, (b) equilibrium equations 55

 

Figure 3.13 - Normalized transverse shear Stresses at the edge of a simply-supported [O/
9010] laminate with S=4 by using difi'erent laminate theories. (a) conStitutive

X

equmons,‘ (b) equilibrium equations 57

Figure 3.14 - Normalized midspan deflections all/2.0) of a simply-supported [0/90] lami-
nate under cylindrical bending by using difi'erent laminate theories .... ...... 63

 

Figure 3.15 - Normalized inplane normal stresses o,.(ll2,lrl2) of a simply-supported [0/90]
laminate undercylindrical bending by using difi‘erent laminatetheories .... 64
Figure 3.16 - Normalized transverse shear stresses on( 0, 0) of a simply-supported [0/90]
laminate under cylindrical bending by using difi'erent laminate theories .... 65

Figure 3.17 -Normalized transverse normal stresses o,( (/2, 0) of a simplymrpported [Ol

90] laminate under cylindrical bending by using difierent laminate theories
66
Figure 4.1 - Cross-sections of original and reduced layups 69

Figure 4.2- Normalized midspan deflections of a [0/90/0/90/0]s laminate under cylindri-
cal bending at difl'erent aspect ratios from difl‘erent layer reducdon approach-
es . -- - 72

Figure 4.3 - Normalized inplane normal Stress 6‘ (1/2, ill/2) of a [0/90/0/90/0]s laminate
under cylindrical bending at difl'erent aspect ratios fi'om different layer re-
duction approaches 73

Figure 4.4 - Normalized transverse shear stress 6“ (0, 0) of a [0/90/0/90/0]S laminate un-
der cylindrical bending at different aspect ratios from difi'erent layer reduc-
tion approaches 74
Figlne 4.5 - Normaliud inplane displacement ti (tr/2. 0) of a square [0/90/0/90/0]s lami-
nate subjected to bidirectional bending at S=4 from difi‘erent layer reduction
approaches 76

Figure 4.6 - Normalized inplane displacement H0, b/2) of a square [0/90/0/90/Ols lami-
nate subjected to bidirectional bending at S=4 from difi‘erent layer reduction

approaches 77
Figlne 4.7 - Normalind inplane normal stress Ego/2. b/2) of a square [0190/0/90/01s
laminate subjected to bidirectional bending at 8-4 hour difi‘erent layer re-
duction approaches - 78

Figure 4.8- Normalind inplane normal Stress 5,(a/2. b/Z) of a square [0190/0/90/0]s
laminate subjecwd to bidirectional bending at S=4 fi'om difi‘erent layer re-
ducrion approaches - 79

Figure 4.9- Normalized inplane shear stress “11(0’ 0) of a square [0/90/0/90/013 laminate
subjected to bidirectional bending at S=4 from difi'erent layer reduction ap~

preaches -80
Figure 4.10 - Normalized transverse shear stress Saw, 1272) of a square [0/9010/90/015

 

 

 

 

 

 

 

 

 

 

xi

laminate subjected to bidirectional bending at S=4 from difi'erent layer re-
ducrion approaches 81

Figtne 4.11 - Normalized transverse shear stress 5"(0/2. 0) of a square [0/90/0/90/013
laminate subjected to bidirectional hearing at S=4 from difi'erent layer re-

 

 

 

duction approaches 32
Figure 5.1 - Fundamental frequencies of simply-supported [01.[0/90] and [0/90/01 lami-
nates with different aspeCt ratios under cylindrical bending 88

Figure 5.2 - Pinned-pinned [0/90] laminate with aspect ratio S=225 subjected to uniformly
distribumd loading : (a) inplane force resultant: (b) midspan deflection 101

Figure 5.3 - Normalized stresses of a pinned-pinned [0/90] laminate with S=225 subject-
ed to uniformly distributed loadings : (a) o; ( llZ,-h/2) ; (b) on ( 0, 0) ..... 103

Figure 5.4 - Normalized stresses of a pinned-pinned [0/90] laminate with S=225 subject-
ed to uniformly distributed loading : (a) (5x ( (/2, 'z) ; (b) on ( 0, z) ........... 104

Figure 5.5 - Normalized nonlinear results of [0/90] laminated beam with S=225 subjeCted
' to uniformly disnibuted loading in three different boundary conditions : (a)

 

midspan deflections; (b) inplane force resultants - 105
Figure 5.6 - ’Ihe load-deflation awe of a square [0] laminate with all edge clamped and
all: a 100 is subjected to uniformly distributed loading 107

 

Figure 5.7 - The load-deflection curve of a simply-supported [0/90] laminate under cylin-
drical bending with S=225 is subjected to inplane compressive loading .. 108

Figure 5.8 - The load-deflection curve of a simply-supported square [0/90] laminate with
alll = 1000 is subjected to inplane compressive loading in the x-direction

 

- 110
Figure 5.9 - ‘lhe amplitude—dependent fundamental frequency of a pinned-pinned [0/90/
9010] laminate with 8:100 under cylindrical bending 112

 

Figure 5.10 - Normalized amplitude-dependent fundamental frequencies of [0190/90/0]
laminate in three difi‘erent boundary conditions : (a) 83100; (I!) 82-10 113

Figure 5.11 - The change of nonlinear fundamental fi'equency and mode shape of [0190/
90/0] laminate at the vibration amplitude Al r 32.0 : (a) fundamental frequen-
cy; (b) coherence factor 115

Figme 5.12 - Mesh layout for [45/4513 laminate in calculation of free-edge stress ...... 1.17

Figure 5.13 -Normalized results of a [45l-45]s laminate subjected to uniform inplane load-
ing : (a) the tluough-the-width in-plane displacement u (0, y, hIZ) ; (b) the
through-the-width interlaminar shear stress on ( 0, y, [214) 118

 

 

CHAPTER 1
INTRODUCTION

1.1 Motivation

. Fiber-reinforced composite materials have been widely used in both aerospace and
automotive industries since 1960 due to their high stiflness—to-weight and high strength-
to-weight ratios. Their flexibilities in design and manufactming are also excellent. How-
ever, because of the heterogeneity of the composite materials through the thickness and
the anisotropy in the individual layers, the design and analytical techniques developed for
conventional materials and structures cannot be used for composite materials. For exam-
ple, it is more accurate to express the strength of a composite material by a curve of prob-
ability of failure instead of a single value; the sums concentration around a cutout in a.
laminated composite must account for the boundary-layer efi‘ect; the low ratiolof trans-
verse shear modulus to inplane tensile modulus renders the composite laminates more vul-
nerable to transverse shear deformation; and the coupling efi'ects among the inplane
loading, inplane shear deformation, and out-of-plane deformation make the prediction of
composite behavior more complicated. All these unconventional phenomena stimulate
new studies on the behavior of composite materials and sn'uctures.

The first theory used in the analysis of laminated composites is the classical lami-
nate theory (CLT). It is based on Kirchhoff's deformation assumptions. However, due to
the low transverse shear modulus of the composite laminates, CLT seems to overestimate
the stifi'ness of laminated composites due to the neglect of transverse shear deformatic‘m.
CLT has been there for long time. In recent years, many investigations have been focused
on the development of new or refined laminate theories to improve the prediction of the
behaviors of laminated composites with various types of geometry and loading conditions

1

[1-4].

2 By modifying the assumption of the displacement field of CLT, the firstoorder
shear deformation theory (FSDT) [7] and the high-order shear deformation theories
(HSDT) [8-12] take the transverse shear deformation into account and therefore improve
the accuracy of composite analysis. Although the properties of the individual layers are
considaedmmesehnunammeones,meyvhmanyneumecompositehminawsassino
gle-layer structures. Generally, these single-layer approaches give good results in global
responses, such as deflection, vibration frequency, critical buckling load, etc. However, as
far as the local responses of the composite laminates are concerned, the single-layer
approaches usually cannot generate satisfactory results. For example, the transverse
Stresses and through-the-thickness deformation cannot be obtained fi‘om these techniques
directly. Unfortunately, these kinds of local information are crucial to the analysis of
delamination, debonding, and flee-edge efi‘ect in composite laminates.

In view of - the problems, a laminate theory based on multiple-layer approach is
really desired. Among the investigations in this area, the generalized laminated plate the-
ory (GLPT) [5], is the most recent and advanced technique. However, since the displace-
ment field used in the GLPT does not satisfy the interlaminar stress continuity at the
composite interfaces, the calculation of transverse Stresses needs to resort to Stress recov-
ery technique which is usually achieved by using equilibrium equations. During the stress
recovery process, the numerical differentiation can worsen the accuracy of the results.
This deficiency can be overcome with the introduction of interlaminar stress continuity on
the composite interfaces. In addition, the incorporation of interlaminar stress continuity
conditions in the displacement field has the potential to increase the accuracy and to
decrease the degree-of-freedom of the GLPT. This motivates the studies canied out in this

thesis.

1.2 literature Review

Structures composed of laminated composites are frequently modeled as single-
layer plates by classical laminate theory. However, as the aspect ratio, i.e., the span to
thickness ratio, of a structure becomes smaller, the CLT can produce errOneous results [6].
Thisisduemmeneglectofnansvasesheardefmmadonwhichisuidcanyimpommm
materials which have relatively low transverse shear modulus compared to inplane tensile
modulus. To account for this deficiency, the idea of Reissner-Mindlin plate theory for iso-
tropic plates was first adopted by Yang, Norris, and Stavsky [7] for composite laminates.
However, the determinatiOn of shear correction factor for some particular problems is very
difficult. This shortcoming of the first-order shear deformation theory was overcome by
the so—called higher order shear deformation. theories [8-12]. The introduction of a higher-
order displacement field made the shear correction factor redundant in the analysis. It also
automatically improved the accmacy of nansverse shear stress disnibution. Although the
high-Order laminate theories gave better predictions of global responses, such as deflec-
tion, vibration fi'equency, and critical buckling. load, they were of single-layer approach
and discounted the independence of individual layers. Hence, the transverse stresses could
not be obtained satisfactorily from constitutive equations.

By considering the layers in a composite laminate individually, the multiple-layer
approaches generally produced more accurate results for both global and local responses.
According to the variational theorems employed, the approaches used for multiple-layer
laminate theories can be divided into the following four categories.

(1) Ambartsumyan’s Approach

This method was first proposed by Ambartsumyan for symmetric cross-ply lami-
nates [13], and further generalized by Whimey for symmetric laminates [14]. In this
approach, a continuous transverse shear stress field was assumed for composite laminates
first. Then, by using consdtutive equations and integration through the thickness, a dis-
placement field was obtained. Based on this displacement field and equations of motion

A

4
from classical plate theory, the governing difi'erential equations were derived for compos-
ite analysis. Since no variational principle was used in this analysis, the displacement
field, governing equations, and boundary conditions obtained from this approach were
variationally inconsistent. Moreover, the solutions only showed small improvements in
the globfl responses.
(2) Hybrid-stress Finite Element Method .

Due to the dificulty in satisfying the transverse stress continuity at the composite
interfaces by using conventional displacement-based finite element method [15], a so-
called hybrid-stress finite element method was developed to overcOme this problem by
assuming a stress field for finite elements [16,17]. with the assumed stress field, which
satisfied the equilibrium equations exactly, and the principle of. minimum complementary
energy, the formulation of finite element analysis was achieved. Because of the carefully
assumed stress field, the stresses resulted fi'om this technique were very accurate when
compared with elasticity solutions. However, the shortcoming of this method was the
sophiStication in determining an appropriate stress field. In addition, as the order of the
stress field increased the derivation becamevery tedious. '

(3) Mixed Variational Principle

Another method to satisfy both displacement and transverse stress continuity con-
ditions at the composite interfaces was to assume displacement field and transverse stress
field independently. This technique was performed by MW and Toledano [18,19]
with the use of a mixed variational principle developed by Reissner [20]. Although the
inplane response was greatly improved by this technique, the transverse stresses needed to
resort to the equilibrium equations for more accurate results.

(4) Principle of Virtual Displacement

In this category, all approaches were based on assumed displacement fields. Seide

[21] assumed a layer-wise linear displacement field for composite laminates and solved

simultaneous equations for individual layers by considering interfacial continuity condi-

5

tions. DiSciuva developed a shear-deformable rectangular plate element based on a piece-
wise linear displacement field [22.23]. thb this linear displacement field, the transverse
shear stresses satisfied the continuity condition at the interfaces of the composite laminate.
However, the shear traction boundary conditions at top and bottom sm'faces of composite
mnfinamswaenmassmedTherefom,meuansversesnessescouldnmbecalculawd
directly from constitutive equations. Another approach, proposed by l-linrichsen and Pala-
zotto [24], used a cubic spline functions to describe the displacement field in the thickness
direction. However, this C2 continuous displacement field resulted in a continuous strain
field through the thickness, hence overconstrained the composite response. Recently, a so-
called generalized laminated plate theory (GLPT) was presented by Reddy [5]. It was fur-
ther expanded by his colleagues [25,26]. In this theory, a layer-wise representation of
inplane displacements resulted in improved inplane response and transverse shear defor-
mation. However, due to the low-order displacement field used, the surface shear traction
boundary conditions and the interfacial transverse shear stress continuities could not be
satisfied beforehand [25,26]. A sophisticated technique using equilibrium equations for
recovering transverse stresses man be enforced in the porn-process calculation [26].

Along with all the attempts mentioned above to solve the response of composite
laminates, there was little success in using elasticity approach. Pagano [6], and Pagano
and Hatfield [27] solved simply-supported cross-ply laminates under cylindrical bending
and bidirectional bending, respectively. The exact solution of natural frequencies for lami-
nates under cylindrical bending was presented by Jones for cross-ply layups [28] and 0&-
axis laminae [29]. Kulkarni and Pagano extended this technique for ofi'-axis laminates
[30]. The vibration analysis for rectangular laminates by Srinivas, Rao, and Rao [31] was
limited to laminates composed of isotropic layers, while the study by Noor was associated
with vibration of crossoply laminates [32] and stability of multi-layered composites [33].
The results from elasticity solutions can serve as examples for assessing the laminate the-

ories.

1.3 Praent Studies

Upon the demand for finding both displacements and stresses accurately and em-
ciently in the composite laminate analysis, it is the intention of this study to develop a dis-
placement-based laminate theory which can calculate the transverse stresses directly from
the constitutive equations. Hence, the numerical difi‘erentiation during the recovery of
transverse stresses, which usually reduces the accuracy of the results, and other deficiency
of the recovery technique for some particular problems [34] can be avoided.

In order to calculate the transverse stresses direcrly from the constitutive equa-
tions, the displacement field should conform to the stress field in the laminate. In other
werds, the continuity of interfacial tractions and the boundary tractions at top and bottom
surfaces of composite laminates need to be satisfied exactly when the displacement field is
assumed. These requirements can be accomplished by assuming layer-wise cubic dis-
placement functions through the thickness and incorporating the traction boundary condi-
tions in the formulation. With this conformal displacement field, the governing equations
and associated boundary conditions can be obtained via the principle of virtual displace-
ment. In this study, an interlaminar stress continuity theory (1301) is derived first. Then,
by assuming constant transverse displacement through the thickness, which is used in
most laminate theories, the derivation can be reduced to interlaminar shear stress continu-
ity theory (ISSCT). The formulations of these theories consritute Chapter 2 of this thesis.

In Chapter 3, numerical examples for static bending are used to demonstrate the
accuracy of these interlaminar Stress continuity theories by comparing them with elasticity
solutions. In addition, a comprehensive disscusion regarding the Stress continuity theories
and other laminate theories is also presented.

Due to the increase of the order of displacement funcrion through the thickness, the
number of displacement variables in the interlaminar stress continuity theories increases
accordingly. As the number of layers in a composite laminate increases dramatically, the
burden of a huge number of degree-of-freedom on the computational efiort can easily

7
jeopardize the feasiblity of these theories. Therefore, a layer reduction technique is pro-
posed in Chapter 4 with a goal to keep the computational efi‘ort to minimum while still
I retain fair accuracy. The demonsn'ation of this technique is canied out for ISSCT only,
though similarprocedluecanbeusedforISCT.

Chapter 5 presents the applications of ISSCT for natural vibration, linear buckling
load,nonlinearbending,nonlinearvibration,andfree-edge stressesofcomposite lami-
nates. Finally, the conclusions and recommendations for this study are summarized in
Chapter 6.

CHAPTER 2
INTERLAMINAR STRESS CONTINUITY THEORIES

2.1 Introduction

Ever since the use of classical laminate theory (CLT), many studies were devoted
to the development of a more accurate theory for composite Stress analysis. First was the
first-order shear deformation theory (FSDT). It accounted for transverse shear deforma-
- tion which was ignored in_the CLT. However, the difiiculty in determining shear correc-
tion factor for FSDT rendered it inconvenient to use. By assuming higher order
displacement field, the high-order shear deformation theories (HSDT) overcame the prob-
lem of shear correction factor. They were also able to give good results for deflection and
vibration analysis. Regardless of their advantages, a more refined theory was desired to
present more accurate stresses. By modeling the individual layers of a composite laminate
separately, the multiple-layer theories gave improvement in predicfing both deflection and
stress state. Nevertheless, among the developed multiple-layer theories, the continuity of
interlaminar Stresses was not satisfied. Hence, the correct transverse stresses could only be
obtained by means of equilibrium equations. In this respect, two interlaminar stress conti-
nuity theories which allow the calculation of transverse stresses directly from the consritu-

tive equations are presented.

2.2 Interlaminar Stress Continuity Theory (ISCT)
In deriving the interlaminar stress continuity theory, the following displacement
field is assumed for an n-layer composite laminate:

u(x.y.z) - Z (U;-j(x.)')¢1(°+7'2i-2(x.y)¢zm+U,-(X.y)¢3m+7‘2i-l(x.y)¢4m)

i-l

v(x.y.z z) - Z (V-,(x.y)¢‘°+32t-t(x.y)¢‘°+v(sy)¢,‘°+$tt-ttx.y)¢§°) (2.1)

i-l

W(x.y.z) . 2 (W-1(I-))¢m+R2i-2(X.J)¢2m+W;(Z.7)45m+ku-l(xel)¢qm)

in!

where o’s are so—called Hermite cubic interpolation functions and are defined as follows

2 2-2- 3
(i) a zr'--l s-l
¢ 1-2-3( hi ) +:(12 hi )

 

 

¢2(0‘ (3“ 3;-1) (1'13.

 

 

 

 

2 3 zi-l 5 z 5 ‘i (23)
"’ "l ' ‘l 3:“ l
..._., MI: )- 1r)
Q1“) 3 ¢2(0 =3 Q3“) 3 ¢:0 3 0 Z<Zi_‘ or 2>Zi

As depicred in Figure 2.1, (i) represents for the number of layer and ’1‘- the thick-
ness of the layer. U... V,, and W“. denom single-valued displacement components at the in-
terface between (i) and (3+1) layers in x, y, and 2 directions, respectively. Hence, the
continuity of displacements across the interface is enforced In addition, t2“, 32'“, and
R25- 2 stand for the firm derivatives of u. v, and iv with respect to 2 immediately below, the 1'-
interface, respectively, while 21,--" 325-1, and Rzi-1 above the interface. Since the conti-
nuity of interlaminar tractions must be satisfied at every interface, some of the first deriva-
tives can be eliminated. .

In this Study, composite laminates are assumed to deform within a linear elaStic

range. Hence, the following linear strain-displacement relations hold.

10

 

\

\W \

v3”
R2, 3 ‘ 3"" (i+1)
w, u
Rat-l if” /
323-2 Tfl-l
vi-l .
WM Its--21) it" (i) h,- 2‘

Ru-s

 

 

i-l

 

. I .
\ 3%

 

i

 

 

 

 

Figlne 2.1 Coordinate system and displacement variables.

e ‘au 8 .3v 8:
8 33' I 5;, t 5?
(2.3)
Bu 3v 3v 3w an 3w
2‘» ' 5+5; ' 2"» ' 5+3; . 2a.. ' #3:

With these main components, the Stresses in each layer can be calculated tom the follow-

ing constitutive equations for orthouopic materials.

 

 

 

 

 

 

, . (a) . . m , . (o
“x Q1! 9:2 Qt: 916 ‘3
4 5, > . Qt: 922 923 925 l ‘1 » (2.4a)
O" 013 Q23 Q33 Q36 :2
. a” . _Ql€ Q“ Q36 956‘ t 23:7 .
0’ (D Q Q (I) 2! (i)
y: . 44 45 ,2
{ct} [e.g.,] {2...} <14”)

where the definitions of Q’s can be found in Reference [35]. By substituting the displace-
ment field into the transverse shear strains in the strain-diaplacement relations and then the
constitutive equations, the continuity condition for the transverse shear stresses at the i-in-

terface can be employed, i.e.,

(0 (H1)

0' 0’
{ a" } . { an } i. 1,23...n-1 (2-5)
3‘ x - x, xx ‘ " ‘1 ‘

These equations can help to eliminate some variables. In fact, the following correlations

between the first derivatives can be eStablished,

aw; .

{32i-1}=[A]m{32i}+[81m 5’- (2.6)
1.25-! 23‘ 3V;
5}

where

12
(i) l l I g
[ A10) . Au‘u . Q” Q4? Q(+l) as H) (27
A A (0 (5) (5+1) (5+1) ° a)
2! 22 Q45 Q55 Q45 9,,

' (0 (3)
(‘3 ,_ 311312 . Au‘tz _ 10 .
I B ] [th 52] L21 0 1 ts 1.2.3....n-1 (2.7b)

Similarly, the continuity condition of the transverse normal Stress at the i-interface,

i.e.,
“ML"; = o“”’| i=12.3.....n-I (23)
can also be satisfied by requiring
an 30,. ‘ av‘. ‘ 3V; I
Rz‘- l 8 C :03}; +C 205i +C‘35; +C3()a-’- +C4()R23 , (2.9)
inwhich
(5+1) (0
cf“: Q” a? '99 (2.10a)
(1'44) (1')
cm. Q” 0 Q” (2.10b)
Q33 l
‘ Q(E+l)_ Q0)
cg.) 8 930) (2.103)
can)
C4“) _33(_)_ (2.10d)
9;;

From Equations (2.6) and (2.9), it is clear that the first derivatives of u, v, and w
with respect to 2 right below the interface are related to those right above the interface and

some other dispacement components. By letting

13

[AW] ’ [3?] and [5“] " [38] (2.11a)

cf" - cz‘" - c§"’ - 0; Ci” - 1 _ (2.1113)

and changing the notations

72.17.- ; SZi’S.’ ; RZi‘R; i=0J23.....n-I

(2.12) .
TZa-l 3 T. ' '

328-1: 5,. ; RZn-l = R.

the displacement field can be rewritten as follows,

" 3W. aw. '
u =- 2 (”i-14’1“) ”ms” + Uni" ”$93; ‘+B§§’¢£°;; ‘+A$¢.“’T.+A2‘i’¢.‘°59

i-l

" ' duh anfl
V " 2 (V'- 1‘91“) +5.2-14’2“) +Vt¢§° ”$293; ‘+Br(?¢4m§; i""41?4’4mTt'*'Ai(?‘l’:°5i)

i-l
. 30- av. 3v.
(0 ‘ ‘ (0') ‘ ' (0 I a
w " Zm'd‘”! +R5’1¢2()+Wi¢;)+cl has; “'2 dub? +§;‘)+

i-l

av‘. ‘ ‘
c§°¢§°$ +C§’¢§’R,) (2.13)

It should be noted that the interlaminar stress continuity conditions reduce the total
number of displacement variables floor 913 + 3 of Equation (2.1)~ to 6» + 6 of Equation .
(2.13). In this study, for simplicity, a bidirectional laminate with dimensions of a xb sub-
jeeted to a disuibuted lateral load q(z, y) on the top surface of the laminate is presented.
The shear tractions on b0th top and battom surfaces and the normal traction on the bottom
surface are all equal to zero. Hence, with the principle of virtual displacement, the follow-

ing variational equation can be written for the composite laminate considered herein,

° '13:

 

NOE

”la.

 

 

Qq ”Q Vq “Q

 

Se

8

5e,

8e

 

 

8
5 (23:) J

14

O":
as:

}T{

5 (22”)
5 (22,.)

}

\

 

J

dz - qGW.

 

4’41 (2.14)

By using the linear Strain-displacement relation, Equation (2.3) can be expressed

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

as
r t (9
ex '
e (i) (0
t , » = (N, H2. } (2.1521)
g:
L 2“, J
28 (0
n} .. [N9] {29} (2.15b)
28:: ’
where the maniacs are defined as follows,
4’1 ¢2 4’3
[Min]. *1 4’2 f ¢3
V1 4"2 CW4 (32¢, 62°} 634’} V3
_ o, ‘91 ¢2 ¢2 ‘13 4’3
32294 32194 A22” A21” -
312% 311% A12” All¢4
- 649.4 (2.168)
B
312% +811” 321% A12” A22¢4 Ari¢4 Azr¢4 -
22%
g z o g I i i u 2 C ¢ {C Q l 0
~ [5 g m H «i an ,. :14
IN" 18 o g 0 ' I g a
M i i i i l it: its 4’2: ts CnMCztt
lcz‘i’ai C3” 3124’} ' Bti¢'4+¢3 A12“ Anni ”4%
i l . g (2.16b)
§C2°4§C3¢4§ 322¢4+¢3 32W) A22¢'4‘ Aar‘P'sicflhi

 

 

 

 

me [avg -Hau ,av.,1 av _,W a’w. _.,a’w 137w, ar._ ar._,as.,as.
:- a; a; 25 W423; 2333; 7,2231; '5; '3';

aumauavav. a’wakv a’whmara 23,3533,
‘2""33 252323;“; $322572 5; a- a; 52‘s]

(0 r a’u._ 320.- a’u. 32V a’v 31th,, ._ aw.-
{x } - U _t l l _t- -l -l l I l T S
: 5-1.3:: 253' '3’; ' Vi-l'- 3:2 'm’ .372 1'}: '5; ’ i-l’ i-l

aRMan air/Hazy azu a7v‘a2v‘ a’vmawaw arias.

a; 2a; 2 ”2372 553272 ”23:2 6'3??? a; a'2 ’32 22a? 5;](2217b)

It should be timed that (') depicts the differentiation with respect to 2. Substituting the
above expressions into Equation (2.15) and using Equation (2.4), the principle of virtual

displacement becomes

0 = jg]; ( {53.}TISknl {1.} + {83.}Ttska {2.} ~45 W.)dydx (2.18)

In the above equation, the following notations, which represent for the assembled
matrices through the thickness, are used.

 

 

{2.} = 2 {2w} (2.19a)
In!
{2.} . 2 {3:0} (2.19b)
it 1
,. - (8')
‘ QuQuQuQm
[$12.] =- 2 j? [NPIT 912 922 923 an t~£°ldz (2.20a)
5.1 M Q13 923 933 Q36
_ Q16 Q26 Q36 QGG‘

16

. (5)
221K” ’ Q45 955 ’ z ( )

It is not dificult to see that {2.} has dimensions of (1313+ 13) x1 while {2,}
(14:: + 14) x 1. Since the laminate surface traction conditions should be satisfied in the as-
sumed displacement field, the number of displacement variables can be further reduced.

First, the vanished shear tractions on the surfaces give rise to

 

 

‘” m (I) s +3W°
a): a Q44 Q45 0 33' s { 0 } (2.213)
On Q (1) Q (1) 3W0 0
x :- z. 45 55 TO + a;
"" ( ) (a) 3W“

“22 . 4: 945 5"”; - {0} 221
o m m aw o ( 2 b)

It 2 _ 8. Q45 955 T. + 5; I

Because the matrices of shear moduli are nonsingular, the following equations can be con-

eluded.

aw0 aw”

so. - a; ; 5,,- -3-; (222a)
aw aw .

r0... -5; °' ; r“. -a.; 2 (2.22b)

Similarly, the conditions to satisfy the applied normal tractions on laminate surfaces,

0'

, 5'0 ; 0, h-qow)

z... z-

2

 

NI

can be achieved with the following two equations,

QS’aU, Qg’avo egg) (at/0+avo)
0" " '2
ngufi ngy 9321) 3'; ‘3}

 

 

(223a)

17

ag’au Q"’av Q“) (“an av '
R22" Q7333 "QT-23'; QT.) (33 M32) Qt») (2223b)

 

By incorporating Equations (2.22a.b) and (2.23a,b) in Equations (2.17a,b), the as-
sembled matrices {2.} and {2,} can be associated with the reduced ones, {3.} and

{2%,} , i.e.,
{3.} - IE.) {22.} + {22,} (2.24a)
{2.} - t5,1{i.}+tq,} (2.2413)

where [5.] and [5,] are constraint matrices with dimensions of (1312+ 13) x (13:: +3)

and (14:: + 14) x (141: + 6) , respectively, while {qn} and {21,} are associated column ma-
triees related to the distributed loading q (x, y) . Details of these matrices are listed in Ap-
pendix A. It can also be concluded that the tOtal number of independent displacement
variables required for the reduced displacement field is 621. Since b0th {qn} and {(1,} are
known quantities, the variation of these two column matrices will vanish. Therefore, the

substitution of Equation (2.24) into Equation (2.18) yields

0 3 ISI;( {aifl}r( [Skill {in} 4" [5.] [SkJ {qn})+

- T . - (2.25)
{am ([SKJ {L} + [5,1 [ska {4m «:5 W,)dydx
where
[312.] =- tEJTtsk.) [5,1 (2.26a)
[31h] . [5,17[SK.1[E,] (2.26b)

2.3 Interlaminar Shear Stress Continuity Theory (ISSCT)
In the foregoing formulation for ISCT, the variation of transverse displacement w

in the thickness direction has been taken into account. For very thick composite laminates.

18

as it will be seen later, this consideration provides a more accurate modeling for composite
deformation and stress analysis. However, the high degree-of-freedom results from this
assumption becomes a major concern for analysis eficiency. Moreover, as the aspect ratio
of a composite laminate increases, the assumption of uniform transverse displacement
through. the thickness becomes more practical. Hence, there is a need to have a simpler
theory for composite lamainate analysis. An interlaminar shear stress continuity theory is
then proposed. .

Following the notations used in the previous derivan'on, the displacement field for

the interlaminar shear stress continuity theory can be written as

u(x.y.z) - 2w.- -,(x.y>¢‘°+th-2(x.y)¢,‘°+v,(x.y)¢§°+th-t<x.y)¢§“)

i-l

V(x.y.z) - 2 (V,-, mm,” +32t-z<2.y)¢‘° +V,(x.y)¢,“’ +32t-t(x.y)¢£°) (227)

hi

Maya) 2.. wo(x.y)

It should be noted that. by assuming constant w over the thickness, the normal
strain in the z direction vanishes. Hence the effect of transverse normal Stress is neglected
in this theory. In ether words, the continuity condition of transverse normal stress, Equa-
tion (2.8), is automatically satisfied. Again, the number of the first derivatives in Equation
(227) can be reduced by employing the continuity conditions of transverse shear stresses
at the composite interfaces, i.e., Equation (2.5). With the same notations used in Equation

(2.12), the reduced displacement field becomes

. aw, a»,
as Zwiqtfhrt- ,¢‘°+U¢‘°+AQ¢,“T+ ‘;’¢‘°S. +B‘°¢‘°5;° +B§?¢‘°;;°)

i-l

aw, , aw,
v a 2 (Vidom+S‘._,¢;°+V¢m+A9¢§°T£+Af?¢(033 ”glam; +a,‘,’¢‘°3—;° )

W 3 W0 (Xv Y) - (2.28)

19
The principle of virtual displacement for the laminate with the same geometry and
loading condition as used in the formulation of ISCT can be obtained from Equation
(2.14) by simply dropping the virtual displacement term corresponding to the transverse
normal strain, the principle then becomes as follows,

1'

5 °8 5" e' T 5mg
0 b 2
o-M. L. ., n, +{:} {n.4,} w <22”
2 on 5(220) ‘_‘

Using Equations (2.3) and (2.28), the strains can be substituted by the displacement vari-

ables shown below,

" (0
at

e, .. tN£°1{X§°} (230a)
23x,
2‘ (5)

{ 21‘} - tN§°1ttf°i (2.301»

where the following matrix definitions are used,

 

 

 

 

 

(a 4’11 ‘5! 4’3; ‘22” ‘21” I
[Na 1" I .1 ¢2 ¢3 A12¢4 A11¢4§
1’1 ’1 ’21 ¢2l i ¢3 ¢3 A12” ‘n‘hi‘ufi ‘21”
822M 321%
Baa4 8“” (2.31a)

312% i Bll¢4 + 522.4 le¢4

Buv‘ 1+Bu¢

V: IA12¢'4;Al1¢'4
i; 1+3224’th 321W

, i . . ‘ (2.31b)
4’ siAzz¢ 4; Azt" 4

 

 

20

{2(5) T an-l an-l aV'-l aV'-l aT'-l aTi-l asi-l asi-l aUt' aUt' avt' 3V5
l ’5; '5; 93'; 05; 0;; 95; '5; 9;; or, 039;;

3r, ar, as, as, a7», 3’», a’wo
’5’5,3'3.¥’m'$f] (2.3%)
. 1' 3w aw
{2:0} ' [vi-l'Vi-l'Ti-l'Si—l'ut’ V9 Tespgovgo] (2.32»

It should be mentioned that although the same matrix notations as those of Iscr,
are used, the contents“ of these matrices are difi’erent. This also applies to the upcoming
derivations. Plug these expressions into Equation (2.29) and utilize the constitutive equa-
tions of Equation (2.4), after integration over the thickness, a similar equation as obtained

in ISCT can be achieved.

0 . jgj;({83.}rtsk.1 {it} + {82,}TISKJ {3,} -q8w°) dydx (2.33)

In the above equation, all the matrices denOte the corresponding assembled ones over the

thickness,i.e.,

{2.} = Z {29} ; {2.} =- 2 {25°} (2.34)

i-l 3.1

(a)

, an 912 at
[ska . 2]” [NW] (2,, Oz; 925 mm”: (2.35a)

' " Q16 925 966

(a)
(3x3) . 2 fm mm] [9“ Q45] (1)1914: (2.35b)

fill 55

The matrices associated with the inplane strains have a dimension of (8:: +11)
while those with transverse shear Strains (4n + 6) . Moreover, the shear traction-free condi-

tions on bath top and bottom surfaces of the composite laminate enforce the displacement

21
variables to be further reduced. This similar manipulation as performed in the derivation
of ISCT results in the following relations.
3 s aw° ' r r aw°
These equations can be employed to eliminate the dependent displacement variables in
{2.} and {2,} by introducing consn'aint matrices [5.] and [5,] ,i.e.,

{2.} - {5.16.} (2.37a)
{is} 3 [E3] {is} (2.37!»

where {3.} and {12,} are the reduced matrices with dimensions of (81: + 3) and (4n+2) ,
respectively. The complete expressions of [ER] and [5,] can also be found in Appendix
A. With these reduced displacement matrices, the equation for the principle of virtual dis-

placement becomes
0 - [3]; < (522,} 7151?.) ii.) + {512,}TtSl-le ti.) -45wo)dydx (2.38)
where
[sin] = [5.1 Tux.) [5.1 (2.3%)
[512.1 - [lavish] [5,] (2,391,)
2.4 Closed-Form Solution

The governing equations and associated boundary conditions for ISCT and ISSCT
can be obtained by substituting the displacement fields, Equations (2.13) and (2.28), into
the corresponding principle of virtual displacement, i.e., Equations (2.14) and (2.29), re-
Spectively, with the use of Gaussian theorem. Since the pmpose of this study is to discuss

and compare the results from difi‘crent theories. these equations are nOt described here. In-

22
stead, the techniques for solving the closed-form solution and finite element analysis are
presented. Since the solution phases for both ISCT and ISSCT are similar. only ISCT is
discussed.
For a laminate with all edges simply-supported and subjected to bidirectionally si-
nusoidal loading. '

- , m , (it)
ML!) 8 UZIWMTMT (2.40)

there exists a closed-form solution. By assuming the following displacement functions,

' -: mum) m.hty
0.:- 2Ufcos-a—srn— T-s zf‘i’cos—sln—

. b I 1 a b
131-1 “-1
- V" , krtx (rt) S - 3’” .1110 My i=0J2....n (2.41)
VP 2 ism-{CWT 1" 2 ism-7°“? j=1.2.3....n-1

u-l 1.1-1

'_,Intx,hty '-u.knx.lur
2.1-l 1.1-l

the boundary conditions on the edges will be satisfied automatically. In the above equa-
tions, the displacement variables wi “-” on top of them represent for the corresponding
displacement amplitudes. Substitute these displacement functions into Equation (2.25)
and carry out the integration over the x—y plane, the governing equation for the unknown
displacement amplitudes can be obtained,

(0") {2"} = {F"} It. I = 1.2.3....0- (2.42)

In Equation (2.42), [0“] is a coefficient matrix with dimensions of 6:: x Ga, {53’}
is a 6nd column matrix and contains all the unknown displacement amplitudes, and

{5"} a 6» x 1 column matrix associated with the external loading q (x, y) .

2.5 Finite Element Solution

For loading types Other than sinusoidal distribution and boundary conditions other
than simple support, a closed-form solution becomes impossible. Hence, finite element so-
lution should be pursued. In finite element formulation, a set of shape function for an ele-
ment on x-y plane is introduced,

ti.) . [v.1 {X}

. (2.43)
at.) - [VA {1:}

where {X} is the nodal column matrix and [y] ‘s are the matrices consisting of the shape
functions and their derivatives. Plugging Equation (2.43) into Equation (2.25) and per-
forming the integrations, the principle of virtual displacement results in the following fi-

nite element equation

[Kl {X} = {F} (2.44)
in which [K] is the stifincss matrix and {F} is the external loading vector, i.e.,

m . [313(le Tlsim [v.1 + W.) Ttsfm lv,l)dydx ' (2.45a)
{F} = {31; ( [v.1 715,171.93.) {4,} + lvytzdrlskd {42mm

43!: {v} 4414: (245")

In the above equation {on} is the interpolation function associated with W. only.

CHAPTER 3
ASSESSMENTS OF THE STRESS CONTINUITY THEORIES

3.1 Introduction .

The stress continuity theories presented in the previous chapter provide a direct
way to calculate the interlaminar stresses from constitutive equations. In this chapter, sev-
eral examples which have exact elasficity solutions are used to demonstrate the accuracy
andfeasibility of these theories. In addition, since the solutions from other laminate theo-
ries such as I-ISDT and cm are also available, it is the objective of this chapter to com-
pare the advantage and disadvantage of the difl‘erent theories.

3.2 Numerical Examples for Stmes Continuity Theories
3.2.1 Laminates under Cylindrical Bending .

For a composite laminate which consists of cross-ply layups and is under cylindri-
cal bending along x-axis, as shown in Figure 3.1, the displacement field becomes indepen-
dent of y-direction. Consequently, the laminate analysis can be reduced to a two-
dimensional problem and is easy to be done. In fact, Pagano [6] has presented an elasticity
solution for this problem. His investigation has long been considered as a standard study
to assess the accuracy of laminate theories. Therefore, a similar routine is performed here.

Thematerialpropertiesusedinthissmdyareexactly thesameasthoseusedin [6],

i.e.,

5, =- 25x lo‘psmt2 a £3 =- 1x lO‘psi;Gu . 613:»ij lo‘ttsm;23 . 02X 10%:
V12 3 v13 2 v23 2 0.25.

The results are presented with the following normalizations.

24

A ' 4(3) - new?)

      

 

 

h

L ,

Figtn'e 3.1 Simply-supported composite laminate under cylindrical bending.

A
OXOxo

 

 

1005,19w ETu _ ox __ o oz
8 ;u a —--—;0’8 3 —;On a —,6 a .—
qol4 Mo ‘10 40 ‘70

The assessments of the interlaminar stress continuity theories are presented in the
following sections.

1. Assessment of ISSCT

The investigation starts fi'om ISSCI'. Tables 3.1 and 3.2 present both closed-form
solutions and finite element results for asymmetric [0/90] andsymmetric [0/90/0] lami-
nates, respecfively. The normalized midspan deflection and transverse shear stress are of
major interest. In the finite element analysis, Hermite cubic interpolation functions are
used for through-the-thiclmess assembly while both cubic and linear interpolation func-
tions are used for inplane assembly. Hence, in an n-layer composite laminate, the number
of degree-ofofreedom for'one element is 8n+4 when using cubic funcfions while 4:: +4
when using linear functions. As shown in Tables 3.1 and 3.2, only the four-layer ones are
presented with both cubic and linear interpolation functions for comparison. It is clear that
the cubic interpolation has faster convergence than the linear one even when they have the
same number of degrees-of-freedom. This conclusion becomes more distinct as the aspect
ratio of the composite laminates increases. Nevertheless, all the finite element results con-
verge to the closed-form solutions as the number of elements increases.

There is no surprise that the transverse displacement converges faster than the
transverse shear stress. This is because that ISSCT is a displacement-based approach. Fur-
thermore, the number of layers used in the analysis does n0t seem to have significant ef-
fect on the results, especially when the laminates have large aspect ratios, say 8820 and
3:40. It should also be noted that due to the assumption of constant transverse displace-
ment through the thickness, bath the transverse displacement and the transverse shear

stress show high deviations from the exact solutions when S=4. As the aspeCt ratio of the

by using ISSCT.‘

27

Table 3.1 Results of a simply~supported [0/90] laminate under cylindrical bending

 

Aspect

ratio

Iggy.
°mfil

FEM

 

No. of elements

 

2

4

10

20

form

 

mg»)

S=4

Cubic
Cubic
Linear
Cubic

4.7977
4.7983
4. 1088
4 .7983

4.7808
4.7818
4.6748
4.7848

4.7786
4.7796
4.7647
4.7798

4.7786
4.7796
4 .7760
4 77.97

4.7785
4 77.97
4 779.7
4 .7797

4.6950

 

5820

Cubic
Cubic
linear
Cubic

2.7122
2.7122
2.1405
2.7122

2.7078
2.7077
2.5718
2.7077

2.7069
2.7069
2.6878
2.7069

2.7069
2.7069
2.7030
2.7069

2.7069
2.7069
2.7069
2.7069

2.7027

 

Cubic
Cubic
Linear
Cubic

2.6430
2.6430
2.0778
2.6430

2.6414
2.6414
2.5051
2.6414

2.6409
2.6409
2.6201
2.6409

2.6409

2.6362
2.6409

2.6408

2.6407
2.6408

2.6398

 

 

an (o, 0)

Cubic
Cubic
Linea
Cubic

0.8995
0.9395
1 069.8
0.9352

0.8555
0.8763
1.07 10
0.8735

0.8531
0.8704
0.9010
0.8666

0.8530
0.8703
0.87 16
0.8664

0. 8530
O. 8703
0. 8703
0.8664

0.9135

 

8820

Cubic
Cubic
Linear
Cubic

6.1093
6.1055
4.8953
6.0885

4.2095
4.2548
5.3319
4.2532

3.9416
3.9503
5.2201
3.9499

3.9344
3.9380
4.6595
3.9373

3.9340
3.9372
3.9372
3.9365

3.9460

 

 

 

5'2
assume other») onus» onus» onus» cusses) 39 "s
"'0

 

Cubic
Cubic
Linear
Cubic

 

13.933
13.996
9. 7482
13.979

8.811!)
8.8347
10.716
8.8314

7.8414
7.9213
11.031
7.9213

7. 9038
7. 8457
10.451
7.8450

 

7.8374
7. 8395
7 .8395
7.8405

 

7.8436

 

 

28

Table 3.2 Results of a simply-supported [0190/0] laminate under cylindrical bending
by using ISSCT.

 

Aspect
rauo

No.0!

Inter.

$5.22

 

No.ofelemeots

 

2

4

6

10

form

 

Cubic
Linear
Cubic

2.9217
2.6596
2.9217

2.9110
2. 8687
2.9110

2.8965
2.9100

2.9062
2.9098

2.9089
2.9097

2.9096
2.9097

 

8:20

. Cubic
Linear
Cubic

0.6197
0.5039
0.6197

0.6179
0.5909
0.6178

0.6177
0 .6176

0.6176
0.6140
0.6176

0.6176
0.6170
0.6176

0.6176
0.6176
0.6176

0.6172

 

Cubic
Linear
Cubic

0.5381
0.4264
0.5381

0.5370
0.5103
0.5370

0.5368
0.5252
0.5368

0.5368
0.5328
0.5368

0.5368
0.5359
0.5368

0.5368
0. 5368
0.5368

0.5367

 

 

an (o, 0)

Cubic
Linear
Cubic

1.4545
1.6571
1.4655

1.4265
1.5449
1.4351

1.4252
1.4917
1.4328

1.4251
1.4523
1.4322

1.4250
1.4323
1.4321

1.4251
1.4251
1.4321

1.4318

 

S-20

Cubic
Linear
Cubic

9 .7495
10. 446
9 .7613

8.8230
9.7329
8.8279

8.7608
9.5946
8.7639

8.7471
9.4 110
8.7495

8 .7452
8.7474

8.745 1
8.7451
8.7472

8.7490

 

 

 

man Obb aau ouu an; as;

 

Cubic
Linear
Cubic

 

23.462
21. 103
23 .479

18.218
19.662
18.228

17.757
19.463
17.761

17.654
19.293
17.656

17. 640
18. 906
17.642

 

17.639
17.639
17.641

 

17.634

 

 

29

laminates increase, good agreements between the ISSCT and the exact analysis are ob-
tained. In addition, it is concluded that the results of the symmetric layup, [0/90/0], show
better accuracy than those of the asymmetric layup. [0/90].

2. Assessment of ISCT

The same examples are investigated by using ISCT and the results are given in Tables 3.3
and 3.4. Comparing the closed-form solutions of 130' with the exact solutions, it is clear
that b0th the transverse displacement and stresses converge as the number of layer increas-
es. However, the number of elements afi‘ects the solutions in a difierent manner. This re-
sult can be viewed from the finite element aspect ratio which is defined as the ratio of the

element length to the element thickness. For composite laminates with high aspect ratios,

AF....__

f a—a—rfl‘

it requires more elements to keep the/mute element aspect ratioclos\e to one. On the con-
trary, for composite laminates with low aspect ratios, it requires fewer elements. Accord-
ingly. there is 39 advantage to use too many elements for. oompositeiaminatewith small
aspect ratio. It then is ’underStandable that the convergences of the transverse displacement
and transverse shear stress are not monotonic for the cases under investigation. In addi—
tion, for symmetric laminates, [0/90/0], the interlaminar normal Stress converges to the ex-
act solution very well. However, for asymmetric laminate, [0/90], this Stress converges
from lower values at S=4, while from higher values at 3:20 and sa4o.

3. Efi'ect of Aspect Ratio .

In order to verify the feasibility of using the new theories for both thin and thick
composite laminates, the normalized deflections at the midplane and midspan are present-
ed in'Figure 3.2. It is in a logarithmic scale with the aspect ratios ranging from four to 200
for all three laminates, [0], [0/90], and [0/90/0]. The results show that b0th stress continu-
ity theories agree with the exact solutions perfectly through the entire span of aspect ratio.
Hence, they can be used for deflection analysis for bath thin and thick composite lami—

nates.

Table 3.3 Results of a simply-supported [0/90] laminate under cylindrical

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

bending by using ISCT.
fit Egg: No. of elemenu %‘ Exact
2 4 10 20
2 4.7266 4.7281 4.7402 4.7349 4.6918
s-4 4 - 4 .7144 4.7002 4.7014 4.7033 4.6950 4.6950
6 4 .7139 4.6984 4.6979 4.6989 4.6952
__ 1 2 2.7080 2.7037 2.7029 2.7030 2.7027
19(5. 0) $220 4 2.7079 2.7036 2.7028 2.7027 2.7027 2.7027
6 2.7079 2.7036 2.7028 2.7027 2.7027
2 2.6418 2.6403 2.6399 2.6398 2.6398
M 4 2.6418 2.6403 2.6398 2.6398 2.6397 2.6398
6 2.6418 2.6403 2.6398 2.6398 2.6399
2 0.9806 0.9936 1.1218 1.1189 0.9055
S-4 4 0.9907 0.9349 0.9781 1.0563 09212 0.9135
6 0.9854 0.9259 0.9352 0.9742 0.9174
2 6.1426 4.2638 4.1143 4.3936 3.9451
6'“ (0, 0) 8-20 4 6.2074 4.2775 3.9688 3.9878 3.9479 3.9460
6 6.1872 42742 3.9631 3.9563 39472
2 14.371 8.9330 8.0049 8.1679 7.8431
8:40 4 14.384 8.9371 8.2500 7.9340 7. 8665 7.8437 7.8436
6 14.356 8.9305 7.9306 7. 8561 7.8448
2 0.6192 0.7629 0.8531 0.8533 0.8468
s-4 4 0.7041 0. 7468 0.7944 0.7956 0. 7947 0.7860
6 0.7436 0.7609 0.7880 0.7890 0. 7887
l 2 1.0821 0.8203 0.8874 0.8889 0.8875
3 (—,0) S-20 4 1.2045 0.8547 0.8320 0.8282 0.8273 0.8180
' 2 6 12507 0.8768 0.8249 0.8216 0.8208
2 3.2649 1.4036 0.9326 0.8955 0.8891
8-40 4 3 .4873 1.4349 0.8764 0.8345 0.8285 0.8193
6 3.5518 1.4559 0.8710 0.8278 0.8220

 

 

31

Table 3.4 Results of a simply-suppomd [0/90/0] laminate under cylindrical

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

bending by using ISCT.
FEM
Aspect No. of Closed-
m 18M No. of elements tam Exact
2 4 6 10 20
534 4 29025 2.8982 29011 29052 2.9078 2.8868 23868
6 2.8992 2.8887 2.8879 2.8878 28880 2.8872
.. l 4 0.6194 0.6176 0.6174 0.6173 0.6173 0.6173
445.0) 5'20 6 0.6194 0.6175 0.6173 0.6173 0.6173 0.6173 0'51”
4 0.5380 0.5369 05368 0.5367 0.5367 0.5367
5'40 6 0.5380 05369 05367 0.5367 0.5367 0.5367 05357
5‘4 4 1.4548 1.4653 15136 1.6001 1.6534 1.4244 ”318
6 1.4643 1.4337 1.4318 1.4341 1.4396 1.4314 -
-- 4 9.7509 8.8301 8.7740 8.7811 8.8976 8.7462
6.. (0' 0) 5'20 6 9.7613 8.8297 8.7661 8.7514 8.7440 8.7480 “49°
'4 23.471 18.224 17.767 17.673 17.706 17.640
M 6 23.485 18.228 17.763 17.660 17.645 17541 ”-543
s-4 4 05000 0.4993 0.4976 0.4985 0.4985 0.4985 0 4988
6 05000 0.4987 0.4987 0.4987 0.4987 0.4987 -
.. l 4 05018 0.5002 0.5002 05001 05001 05001
“.(5-01 5'20 6 05018 0.5000 0.5000 05001 05001 05001 05001
4 0.5018 0.4997 0.4999 05000 05000 05000
5"” 6 0.5018 0.4997 0.4999 05000 05000 05000 050‘”

 

 

32

 

 

   

 

 

 

 

5 j r I I I I I r I ' I ' T

. - Exact

--- ISCT
- ISSCT
4'4 —0
3-4 ..
13
2d _
[0/90/01

1 - ..
o I r r I ' I I ' I ' I r I ' I

4 6 8 10 20 4-0 60 80100 200

S (t/h) 4

Figure 3.2 Normalized transverse displacements at midspan for [0]. [0/90], and
[0/90/0] laminates with different aspect ratios. '

33

4. Through-The-Thickness Disnibution

Aside from the results calculated at the particular points as discussed in the previ-
ous tables, anather way to assess the results obtained from the new theories is to compare
them with exact solutions through the thickness. The closed-form solutions for [0/90] and
[019010] laminates with S=4 using four-layer analysis are presented in Figures 3.3 to 3.10.
_ In Figures 3.6 and 3.10, the results of transverse normal stress 03 of ISSCT are recovered
from equilibrium equations. Details of the stress recovery technique will be discussed in
Section 3.3.2. From these figures, it is concluded that both ISSCT and ISCT agree with the
elasticity analysis very well. In most cases, ISCT can improve the results of ISSCT slight-
ly except for the transverse normal stress in the [0/90/0] laminate, 'shown in Figure 3.10.
Studies show that the recovered transverse normal Stress from ISSCT gives better predic-
tion than those directly calculated from ISCT.

3.2.2 Laminates under Bidirectional Bending
Consider a simply-supported cross-ply laminate which has dimensions of a x b and
is subjected to bidirectionally sinusoidal loading, i.e., '

. fl . “I
900) 903137937;

as shown in Figure 3.11. The three—dimensional elasticity solution has been obtained by
Pagano and Hatfield [27]. It is the pin-pose of this study to compare the stress continuity
theories with the exact solution. Although the results presented here are all for symmetric
cross-ply layups, the theories have no dimculty in analyzing asymmetric or angle—ply lame

The material properties studied here are exactly the same as those used in cylindri-
cal bending. Sirrrilarly, the following normalized quantifies are used in the presentations

_ 1005,1314 __ - - 12 ._ - I.
w s 400‘ ; (01,432,435) 2 (7;;(0‘, of a”) , (64.65) a 35(0’1'6“)

 

 

— Exact
--- ISCT
ISSCT

 

 

[0/90] "-4‘
Sad.

 

 

 

 

Figure 3.3 Normalized inplane displacement ii (0) of a simply-supporwd [0190]
laminate with S=4 under cylindrical bending.

35

 

I — Exact
---ISCT
0.41 m. ISSCT
0.3-
z/h
0.2-'
0.14

 

 

[0/901
5-4

 

 

 

 

 

Figure3.4Normalizedinlanenormalstress61/2 f ‘ 1-
laminatewithS=4under£ylindricalbending x( )0 asrmpysupported[O/90]

36

 

 

0.5
— Exact
--. ISCT
0.4 4 ISSCT
0.3 -
0.2 -
0.1 4
5
[0/ 90]
S=4

 

 

 

 

Figure 3.5 Normalized transverse shear stress 5,, (0) of a simply-supported [0/90]
laminate with S=4 under cylindrical bending.

37

 

 

0.5
- Exact
--- user
0-4‘ Isscr
0.3a
0.24
0.1«-
110 -
[0/90]
s-4

 

 

 

 

Figure 3. 6 Normalized transverse normal stress a (1/2) of a simply-supported [0190]
laminate with S=4 under cylindrical bending.

38

 

 

 

— Exact
--- user
0.4- . [sscr
Z/h
3ft) ' --5 0:5 ' 1f0 -

6

[0/90/01 "°“

5-4

 

 

 

 

Figtn'e 3.7 Normalized inplane displacement 17 (0) of a simply-supported [0/90/0]
laminate with S=4 under cylindrical bending.

39

 

 

— Exact
--- ISCT
.... (ssc'r 04-4
0.3-
z/h
0.1-
T -'20 - -'10 T 1'0 20
-1- ”x
[0/90/01
s-4

 

 

 

 

 

Figure 3.8 Normalized inplane normal stress 6, (1/2) of a simply-supported [0/90/0]
larrrinate with 8:4 under cylindrical bending.

 

 

 

 

0.5 -
- Exact
--- ISCT
0.4‘ .... ISSCT
0.3 -+
1
0.2 ..
1
1:
0.1 -
- e
r
2/h 0.0 e . - r
1 2
" 1 ‘ dxz :
i
-2 J
t
(r
.53 a
.54 .
[0/90/01
5-4
.55 .

 

 

Figtn'e 3. 9 Normalized transverse shear stress an (0) of a simply-supported [0/90/0]
laminate with S=4 under cylindrical bending.

41

 

 

[0/90/0]
s-4

 

 

 

 

Figure 3.10 Normalized transverse normal stress a (1/2) of a simply-supported [0/90/0]
laminnte with S=4 under cylindrical bending.

42

 

40:.» =- qosmfi-‘lsint-gl '
‘ XV b
y

. 1"1’11! i'lllhk

 

 

/

x ' all edges are simply-supported

Figure 3.11 Simply-supported composite laminate under bidirectionally sinusoidal
loading.

43
1 Since the maximum values of the displacement and Stresses are of major concern. only the
maximum values in the individual cases are reported. The strrdies are based on bath
closed-form solution and finite element analysis and are presented below.
1. Efi'ect of Layer Number

The closed-form solutions of a square (a=b) [WM] and a rectangular (b=3a)
- [0/90/0] laminates from rsscr are given-in Tables 3.5 and 3.6, respectively. Comparing
the results, it is obvious that the increase of layer number introduces very small improve-
ments to b0th displacement and stresses. This is also true for the analysis using ISCT
shown in Tables 3.7 and 3.8.
2. Efi'ect of Laminate Thickness

From Table 3.5 to Table 3.8, it is clear that the conclusion drawn from the result of
composite laminates under cylindrical bending is still valid in the case of bidirectional
bending. More specifically, both IS SCT and ISCT are feasible for the analysis of both thin
and thick composite laminates.
3. Efi'ect of Element Number

In the finite element analysis, because of the symmetry of the problem, only one
quarter of the laminate needs to be examined. Table 3.9 presents the ISSCT results of a
simply-supported [0/90/90/0] square laminate under bidirectional bending. In order to
evaluate the influence of the order of interpolation functions on the inplane assembly, bath
linear and cubic interpolation functions are used. However, the assembly of the displace-
ment components through the laminate thickness uses a 12-term cubic interpolation func-
tion. Based on these interpolation functions for element assembly, the degree-of-freedom
of a four-node rectangular element is 16:: +12 for linear function while 48:: + 12 for cubic.
For a [0/90/90/0] composite laminate, if n a 4 is selected, the total number of degreeoof-
freedom for a finite element mesh can be calculated accordingly. The numbers for differ-
ent cases are presented as with parentheses in Table 3.9. It then can be seen that with ap-

proximately the same number of degree-of—freedom used in the finite element analysis,

 

 

 

 

 

 

 

 

 

 

Table 3.5 Closed-form solutions of a simply-supported [0/90/90/0] square
(asb)laminateunderbidirectional bending by using ISSCI‘.
a s . - - - - .. -
I oluuou 19 a1 62 up a, 66
4-layer 19555 $07048 $06703 02876 02187 $00465
8-layer 1.9555 $07035 $06695 02915 02192 40.0465
4 12-layer 1.9555 $0.7034 20.6694 02918 02193 $00465
+0.720 +0.663 -0.0465
Exact 1.937 -0.684 .0566 0292 0219 +0.04”
4-layer 0.7324 r05583 $03999 0.1957 0.3006 40.0274
8Mm 0fl%:MfilflMEOflm mmrwmu
1° lZ-layer 0.7324 $05583 $03999 0.1962 0.3007 40.0274
+0559 +0.401 00275
M 00737 . 0.559 . 0.403 001% 0.301 + 0.m76
4mm‘0mnrmm1wmnonm wnzwma
8-layer 05078 $05411 $03078 0.1564 0.3272 40.0228
m ”Mm 0mw$umrmwn0fia mnrmmx
+0543 +0.308 00230
Exact 0513 .0543 .0309 0.156 0.328 +0.02”
4-layer 0.4296 $05366 $02699 0.1400 03377 40.0211
8-layer 0.4307 r05380 $02705 0.1398 0.3378 4:0.0212
10° 12.1ayer 0.4318 $05394 $02713 0.1394 0.3381 40.0212
+0539 +0271 .0.0214
Exact 0.435 .0539 .0271 0.139 0339 +0.02“
- - h - - h - - a
6‘ a 04(5' £015), 62 3 a:(‘;’: £913) 9 0‘ a 66(orori'z’)

asagum.§=§m%m

Table 3.6 Closed-form solutions of a simply-supported [0/90/0] rectangular

45

(3a=b) laminate under bidirectional bending by using ISSCT.

 

 

 

 

 

 

 

 

 

 

£- 801 . E 61 62 64 6:5 66
4-layer 28406 $1.1254 $01115 0.0319 05494 400277
6-layer 28409 $1.1209 $01115 0.0320 0.3512 :00277
4 10-layer 2.8409 $1.1205 $01115 0.0320 03512 $00277
+1.140 +0109 -0.0269
Exact 2.820 “00 -0119 0.0334 0.351 +0028,
4-layer 09178 $07265 $00424 0.0154 0.4196 40.0121
6-layer 09178 $07264 $00424 0.0154 0.4199 40.0121
1° 10-layer 09178 $07264 $00424 0.0154 0.4198 40.0121
+0726 +0.0418 -0.0120
Exact 0919 4,725 .004” 0.0152 0.420 +0.01%
4-layer 05073 $06501 $00301 0.0124 04342 400092
6-layer 05073 $05501 $00301 0.0124 0.4343 400092
2° 10-layer 05073 $05501 $00301 0.0120 0.4340 :0.0092
+ 0.650 +0.0294 - 0.0093
Exact 0510 .0650 .00299 0.0119 0.434 +000”
4-layer 05053 $06243 $00260 0.0114 0.4392 $00083
6-layer 05061 $05252 $00260 0.0114 0.4392 40.0083
10° lO-layer 05061 $06253 $00260 0.0109 0.4389 $00083
+0524 +0.0253 - 0.0083
Em‘ om . 0.624 - o.ms3 0.0108 0.439 + 0.m83
- .. a b h - - a b h .. .. h
0'1 3 61(2'2't2)’ .02 3 62(59-2'9i3) 9 0'5 3 0’5 (0- Gotta)

64 =- 6,(;.00) . 6, =- 6,(0,§.0)

Table 3.7 Closed-form solutions of a simplyosupported [0/90/90/0] square

46

(a=b) laminate under bidirectional bending by using ISCT.

 

2
It

Solution

1'17

 

 

 

 

 

61 52 E54 55 c'6
+0.7215 +0.5542 - 0.0457
4-layer 19377 0. .056.“ 02876 02189 “0459
10.7203 + 0.5528 - 0.0457
4 8-layer 1.9369 0.6843 .0665, 02912 02193 +0.04”
+0.720 + 0.553 - 0.0455
Exact 1937 0.684 -0666 0292 0219 ”10458
+ 0.5587 + 0.4010 - 0.0275
+ 0.5585 + 0.4010 - 0.0275
10 8-layer 0.7370 . 0 5591 -0.4026 0.1959 03014 +0.02%
+ 0.559 + 0.401 - 0.0275
40.5428 +03084 -0.0230
4-layer 05130 4,5432 . 0 3088 0.1555 0.3281 +0023,
+ 0.5428 + 0.3084 - 0.0230
+ 0543 + 0.308 - 0.0230
mt 0.513 . 0543 . 0.309 0.156 0.328 + 0.m30
448” 0.4345 10.5389 #03710 0.1389 0.3388 $00214
100 8-layer 0.4345 10.5389 10.2710 0.1389 0.3388 470.0214
Exact 0.435 $0.539 $0.271 0.139 0.339 . $00214

 

 

 

aaaéam.§-am;m

.. ab It .. .. ab It .. .. II
01 . 01(50 E'ti), oz . 02(59 59:2) 9 06 . 05(09 out-2')

 

Table 3. 8 Closed-form solutions of a simply-supported [0190/0] rectangular

47

(3a=b) laminate under bidirectional bending by using ISCT.

 

 

 

 

 

 

 

 

 

 

1'3 sand“ 5 61 6'2 64 6, 50
+1.14% +0.1085 ’ -0.0269
“‘7“ 13215 4.1041 -0.1191 ”332 03494 +00231
+1.1449 +0.1088 -0.0269
4 6-layer 2.8212 «1.0998 -0.1193 0.0333 0.3511 + 0.0281
+1.140 +0.109 -0.0269
Exact 2.820 -1. 100 -0.119 0.0334 0.351 +0.0281
+0.7261 +0.0417 -0.0120
4-layet 0.9189 _0.7256 $0435 0.0152 0.4198 +0013
+0.7260 +0.0418 -0.0120
10 6-layer 0.9189 $3254 $0435 0.0152 0.4201 ”3.0123
+0.726 +0.0418 -0.0120
Exact 0.919 -0.725 .004” 0.0152 0.420 “10123
+0.6500 +0.0294 -0. 0092
4'13,“ 0.5095 . 0.6502 . 0.0299 0.01 19 0.4343 + 0 ”3
406500 +0.0294 - 0.0092
+0.650 +0.0294 0.0093
_ Exact 0.610 -0.650 . 0.0299 0.0119 0.434 +0 (X193
4-layer 0.5077 $06244 $0.0253 0.0108 0.4393 #00083
100 6olayet 0.5077 20.6244 £007.53 0.0108 0.4393 $00083
Exact 0.508 $0.624 £00253 0.0108 0.439 $00083
. - a b 1:" - ab 1h .. .. h
C" .C:(§, -2-, Li) 0': 332(2'3'13)’ 06306(Oooui§)
' 0:6‘(§,0,0). 65:6,(0,g, 0)

48

Table 3.9 Finite element solutions of a simply-supported [0/90/90/0] square

(a=b) laminate under bidirectional bending by using ISSCT.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 No. of Interpolation - - - - - .—
[1 elements function “' ‘1 (’2 °4~ “5 °6
11 mgarOQ' 1.6979 05420 05136 0.3223 02498 0.0367

* Cubzc(204) 1.9484 0.8039 0.7593 03241 02318 0.0463

m 140444171) 19059 0.6748 0.6404 03132 02353 0.0448
Cub1c(459) 1.9516 0.7149 0.6795 02908 02197 0.0464

4 3x3 116641024) 1.9361 0.6933- 0.6584 0.3084 0.2288 0.0459
Cub1c(816) 1.9527 0.7072 0.6723 02879 02186 0.0465

4x4 1.0104475) 1.9457 0.6990 0.6641 0.3036 02253 0.0462
Cubnc(1275) 1.9537 0.7065 0.6709 02876 02186 0.0465

Closed-{01m 1.9555 0.7048 0.6703 02876 02187 0.0465

m M6) 0.5651 03899 0.2921 0.0853 03022 0.0198
00644204) 0.7204 0.6607 0.5032 02766 03306 0.0267

m M171) 0.7043 05281 0.3823 0.1764 0.3189 0.0261
00616059) 0.7309 05727 0.4121 0.2052 03047 0.0275

10 313 M324) 0.7211 05463 0.3929 0.1969 03175 0.0269
Cub1C(816) 0.7315 05610 0.4024 0.1973 03013 0.0274

4x4 1.4494475) 0.7265 05520 0.3963 0.2039 03154 0.0271
0015160275) 0.7319 05605 0.4007 0.1961 03008 0.0274

Closed-{01m 0.7324 05583 03999 0.1957 0.3006 0.0274

In LW“) 0.3107 03122 0.1815 0.2361 01827 0.0133
Cubnc<204) 0.4805 0.6501 0.4460 02775 04069 0.0203

2112 M171) 0.4796 05069 0.2893 0.0343 03116 0.0214
Cubnc<459> 05062 05623 0.3277 0.1836 03413 0.0229

20 3x3 1.0444324) 0.4968 05277 0.3002 0.1077 03317 0.0223
Cub1c(816) 0.5071 05487 0.3134 0.1620 03300 0.0228

4.4 W475) 05020 05340 0.3035 0.1355 03384 0.0225
Cubsc<1275) 0.5074 05461 0.3098 0.1580 03280 0.0228

016660161111 05078 05411 03071 0.1563 03272 0.0228

m 1.1125106) 0.0403 0.0489 00246 13249 04703 0.0019
00014204) 02299 05089 0.3066 0.9368 01338 0.0060

21.2 M171) 0.3749 0.4053 02039 2.3455 0.6838 0.0160
Cub1c(459) 0.4177 05977 0.3170 0.2904 05130 0.0204

100 3x3 Limrmts) 04027 05029 0.2529 12435 01987 0.0198
Cub1c(816) 0.4276 05615 0.2913 0.1933 03940 0.0210

4.4 M475) 0.4187 05230 0.2630 0.6719 00335 0.0206
Cubxc(1275) 0.4291 05471 0.2805 0.1699 03597 0.0211

Closed- 1am 0.4296 05366 0.2699 01400 03377 0.0211

 

 

 

‘ numbers in parenthesis denote the corresponding degree-of-freedom for each mesh

 

49

elements with cubic interpolations show faster convergence at small aspect ratios. Howev-
er, if the aspect ratio of the composite laminate becomes large, although cubic ones still
have better results for transverse stresses, linear ones can give good predictions for inplane
Stresses.Inaddidomitisalsointerestingtoknowthatastheaspectratioofthelaminate
increases, more elements are required for convergence. Because of the excellent results
from ISSCT and the requirement of a very large degree-of-freedom for ISCT, the finite el-
ement analysis based on ISCT is omitted.

3.3.,Comparison of Din‘erent Laminate Theories
In additionto the stress continuity theories, a couple of other laminate theories also
deserve some attention. The comparison of the laminate theories with the stress continuity ‘

theories becomes an important study in assessing ISSCT and ISCT.

3.3.1 Number of Degree-of-Freedom

Although the accuracy is an essential requirement for a good theory, the number of
displacement variables used in the theory can afi'ect the feasibility of the theory. Form-
nately, the rapid renovation of computer has made the computational work easier and fast-
er than ever before. Nevertheless, the reduction in the computational efl‘ort should never
be ignored. In the following sections, two typical displacement-based laminate theories
are compared with the streSs continuity theories for feasibility evaluation. The ErSt one
Stands for a stagrollayor approach. It is, in fact, a high-order shear defamation theory and
has the following displacement field [11]: .

4 z 3 319°
u(x.)'.z) - uo(x.y)+2(V,-§(;) (11,455”

4 z 2 a”
Nam) = vo(x.y)+z(V,-§(;) (154-50)) (3.1)

"(1.751) 3 190(3))

50
In Equation (3.1), 14., v, and w. denote the displacements on the midplane in thex, y, and z
coordinates, respectively, while \y, and v, are the rotations of the normals to the midplane
about y and x axes, respeCtively. It can be seen that the number of displacement variables
is five regardless of the number of layers in the composite laminate.
The other approach is the generalized laminated plate theory [5]. This mutliple-
layer approach has the following displacement field:

N
Mann) - 4,045) + Z (flung-(z)
1"!

N
v(x.y.z) - v00.» + ZW’ 0:.» 01.12) (3.2)
i-1

W(x.y.z) = water)

where the quantifies with subscript 0 denote the midplane displacements, while (05’s are
the global interpolation functions for thickness assembly. (fl and V’ are the nodal displace:
ments relative to the midplane. It should be mad that the number of displacement vari--
ables used in Equation (3.2) totally depends on the order of the interpolation funcfions and
the number of layers, It, in the laminate of interest. For instance, the order of a piecewise
linear interpolation gives rise to N a n + 1. Since UI’ and annish on the midplane accord-
ing to definition, it then results in N . :1. Hence the total number of displacement variables
for linear interpolation is 211 +1. With higher-order interpolation in each layer, the condi-
tions for free shear tractions on tap and bouom surfaces of the composite laminate can
eliminate four more variables. This brings the total number of displacement variables to
4» - 1 and 671 - 1 for quadratic and cubic interpolations, respeCtively.

Table 3.10 gives the comparison of the degree-of-freedom among the theories ‘of
single-layer approach, multiple-layer approach, and the stress continuity theories present-
ed in this thesis. In this table, GLP'I" represents for GLPT bmed on quadratic interpolation
function while GIJ’T’z cubic interpolation function. It can be seen ISSCT and GLP'I'l have

51

Table 3.10 Comparison of different laminate theories for an n-layer laminate.

 

 

 

 

 

ram mm 3:11.614. 13;: aims"... m
HSDT 5 ' ' constitutive equilibrium equilibrium
GLPT l 4n-l constitutive equilibrium equilibrium
GLPTZ 611-1 constitutive equilibrium equilibrium
ISSCT 4114-1 ‘ constitutive constitutive equilibrium

 

 

 

 

ISCT 6n constitutive constitutive constitutive

 

 

 

 

52
nearly the same number of displacement variables. However, ISSCT can describe a cubic
displacement field through the thickness of each layer while GLP’I‘l only quadratic. More-
over, for the same cubic interpolation functions through each layer, (31.1"?z requires nearly
50% more displacement variables than ISSCT. As also shown in Table 3.10, ISCT de-
mands much more displacement variables than ISSCI', though the former is rewarded with
the simplicity of calculating the transverse normal sness directly from constitutive equa-

tions.

3.3.2 Recovery of Transverse Stresses
“ The recovery of transverse stresses from inplane Stresses can be accomplished by

using the equilibrium equations in the absence of body forces, i.e.,

. 04: 8o
on . “1.3 (33" + 5’14: (3.3a)
2
30 30’
an . “It. (fixi’gq) dz (33b)
.5
30 36
O" 8 .120 (538+3’8) dz (34k)
-5

It must be mentioned that in the analysis of closed-form solution, the disnibution of all un-
known variables in the x-y plane are exact functions. In Other words, there is no inplane as-
sembly in the closed-form analysis. Hence, the derivatives involved in Equations (3.3) do
n0t include any error due to numerical difierentiation. However, in the finite element anal-
ysis, a composite laminate is discretized into many elements. The variables need to be as-
sembled. by interpolation functions and are not exact. The errors from numerical
differentiations become unavoidable and always cause losses of accuracy. In additidn,
each integration in Equations (3.3) provides. an integration constant'to be determined by
the boundary conditions. Usually one undetermined constant cann0t satisfy the two trac-

tion boundary conditions at the top and bortom surfaces of the composite lamimte. How-

53
ever, as the finite element result converges to closed-form solution, both traction boundary

conditions can be satisfied.

3.3.3 Closed-Form Solutions for Different Laminate Theories

The basic difi'erence of the laminate theories mentioned in Section 3.3.1 is due to
the assumption of the displacement field through the thickness. Since the closed-form so-
lution is based on a complete function for inplane deformation instead of section-by-sece
tion assembly, it then does nOt introduce error due to approximation and assembly. A
direct insight into the different theories is possible. Therefore, closed-form solutions are
performed for comparing the different laminate theories.

Table 3.11 presents the results of a [0/90] laminate under cylindrical bending.For
HSDT, GLPT‘and GLP'I", since the transverse shear stresses calculated from the constitu-
tive equations are not continuous across the laminate interface, two numerical values, one -
for the layer above the interface and the Other below the interface, are reported in the table.
The transverse shear stress disnibutions through the thickness for some of these theories
are shown in Figure 3.12(a). In addition, the continuous transverse shear Stress distribu-
tions from equilibrium equations can be found in Figure 3.12(b). Because the results from
ISCT and elasticity are very close to each other and so are GLP'I‘2 and ISSCT, the results
from ISCT and Gl..l"l'2 are nor presented in these figures for clearity. Similar results for [0/
9019010] laminate are presented in Table 3.12, Figtn'es 3.130), and (b). it should be noted
that because of symmetric layup, the transverse shear stress at the midplane calculated by
HSDT, GLPT‘, and GLP'I" has only one .value.

The comparison for different laminate theories can be addressed from the follow-
in g viewpoints.

1. Transverse Deflection at Midspan

Comparing the results from the difi'erent theories, it is clear that ISCT gives the

besr predicdon ( error < 0.1% ) of the transverse deflection at the midspan for the aspect

Table 3.11 Closed-form solutions of a [0/90] laminate under cylindrical bending

 

 

 

 

 

 

 

~ by using different laminate theories.
“is m w a; a. a.
3801‘ 4.4445 33.6062 2.4769 0.7147 0.8320
3.0915 0.9907
Gm“ 4.2625 -33.4930 3.2215 0.7411 08220
. 32913 0.8623
611'!" 4.7785 61.0802 0.8479 0.8623 0.7957
5.4 3.7159 0.8553
ISSCT 4.7785 81.0844 0.8530 -—-- 0.7897
3.7158
ISCI‘ 46918 302907 0.9055 ........ 08468"
3.8362
Elasticity 4.6953 -30.0293 0.9135 ---- 0.7860
' 3.8359 ~
HSDT 2.6933 -703.437 12.625 3.8998 0.8202
75.805 5.0501
61.?!“ 2.6867 4703.315 16.611 3.9053 0.8198
76.016 4.4181
GLP‘I'2 2.7069 -700.464 3.9342 3.9353 0.8184
5320 76.563 3.9340
ISSCT 2.7069 -700.459 3.9340 ....... 0.8182
76.562
ISCT 2.7027 699.737 3.9451 «- 08875‘
76.652
Elasticity 2.7027 699.734 3.9460 ---- 0.8180
76.653
HSDT 2.6375 4796.29 25.266 7.8204 0.8198
303.03 10.106
6131“ 2.6359 2796.41 33.257 7.8241 08198
303.26 8.8432
(11.1712 2.6409 279339 7.8380 7.8387 03194
3.40 303.80 7.8375
13801 2.6408 2793.27 78373 -—~- 03193
303.79
ISCT 2.6398 -2792.56 7.8430 ....... 0.88917
303.88 .
Elasticity 2.6398 4792.59 7.8436 ....... 0.8193
303.88

 

 

1. - .. - - 1
oz: egg-.15) ;on-O,,(0.0) ; O,= 645.0)

"' calculated from constitutive equations directly

2/ h

Figure

55

 

 

 

 

 

 

(a)

 

— Exact
-- lSSCT(eonat.)
-- GLPT‘
.... HSDT

 

 

 

 

3.12 Normalized transverse shear stresses at the edge of a simpl}

(b)
y-supported

 

[0/90] laminate with S=4 by using difi'erent laminate theories. (a) constitutive

equations, (0) equilibrium equations.

 

56

Table 3.12 Closed-form solutions of a [0190/9010] laminate under cylindrical bending
by using different laminate theories.

 

 

 

 

 

“33‘ Thea? 1? a; a; 5.. a,
nsnr 32020 $186276 13636 15061 0.7960
GLPT' 33325 4205875 1.4383 1.4400 0.7834
G!!!" 33581 42199049 1.4541 1.4541 0.7862
5'4 ISSCT 33581 $199125 1.4512 --—~ 0.7855
rscr 33360 «19.7062 1.4532 ....... 07776"
202398
Elasticity 33361 .196700 1.4560 ---—- 0.7858
202020
HSDT 0.6885 $284205 7.0885 82173 0.8213
61.?!“ 06796 4287.302 82132 8.1964 0.8206
01218 06797 $287079 8.1973 8.1974 . 0.8206
5‘20 ISSCT 06797 $287080 8.1966 ---— 0.8206
rscr 06793 -287.109 8.1977 -—-- 08185"
286.913
Elasticity 0.6793 -287.108 8.1983 ---- 0.8207
286912
HSDT 05861 $111321 14.195 16.479 0.8222
61.?!" 05889 $1116.35 16504 16.469 0.8220
curt“ 0.5889 $1116.14 16.469 16.469 0.8220
5'40 tsscr 05889 $1116.15 16.469 ---- 08220
rscr 05888 -1116.18 16.469 --- 08201-
111595
Elasticity 05889 -1116.18 16.470 ---- 08220
1115.96

 

 

 

6‘ " 63(éfig) 3 an ' 342(0'0) 3 at ' 5:(-é.0)

‘ calculated from constitutive equations

 

57

 

Z/h 0.0 . ,2 3.

 

 

 

 

 

 

[0/90/01
S=4»

 

 

(a)

 

 

 

— Exact
- - lSSCT(conet.)
- - GLPT'
HSDT
0.1 ~ :
I :
2/h 0.0 r . : E r .
1 1 2
-4 ‘ 1 5x2
-.2- ‘ '
[0/90/0]
S=4-

 

 

 

 

(b)

Figure 3.13 Normalized transverse shear sn'esses at the edge of a simply-supported
[0190/0] laminate with S=4 by using difi'erent laminate theories. (a) constitutive

equations. (b) equilibrium equations.

58

ratios considered However, there is no surprise to see that at large aspect ratio, HSDT can
also give excellent result.
2. Inplane Stress

The inplane stress considered in cylindrical bending is the normal Stress in x-direc-
tion, 6,. Although ISCT again shows the best result ( error < 1% ), the remaining theories
can give accuracy within 0.5% at 8:40. However, the error becomes very large, e.g., 12%,
at S=4.
3. Transverse Shear Stress .
For the transverse shear Stress. ISSCT and GLP'l‘2 have the same accuracy as those of
ISCT except for a [0/90] laminate with S=4. However, since GLP'I‘2 has higher degree-of-
freedom than ISSCT and its transverse shear stress has to be calculated from equilibrium
equations, GLPT 3 is not as eficient as ISSCT. Although GLP'I'l has approximately the
same degree-of-freedom as ISSCT, its result is n0t as good as ISSCI‘. In addition, from
Figure 3.13(b), it is found that HSDT does nor predicr the correcr trend of the n'ansverse
shear stress through the thickness when S=4.
4. Transverse Normal Stress

Among the laminate theories discussed in this study, only ISCT can calculate the
transverse normal Stress directly from the constimtive equations. However, it is surprising
to see that the results obtained by ISCT does nor provide better accuracy than those recov-
ered from equilibrium equations, especially for the asymmetric layup, [0/90] laminate. Al-
though the accuracy can be improvedby increasing the numberoflayers in the analysis as
shown in Table 3.3. 11:: penalty of increasing 1116 degree-Of-fi'eedom may drastically over-
whelm the support of using ISCT.
5. Bidirectional Bending .

Beside the examples for laminates under cylindrical bending, laminates under bidi-
rectional bending are also Studied here. Since there is no advantage of using GLPT 2,
GLP'I'2 is omitted in the following discussion. Thus, GLPT in the following tables denotes

59
the GLPT ‘. The closed-form solutions of a square [0/90/9010] laminate and a rectangular
[0/90/0] laminate are presented in Tables 3.13 and 3.14, respectively. For the transverse
shear stresses, the results obtained from equilibrium equations are reported within paren-
theses right under the quantities calculated directly from the constitiutive equations. More-
over, due to the symmetric stacking sequence of the laminates Studied. only one value of
transverse shear stress is found at the midplane.With the results shown in Tables 3.13 and
3.14, it can be seen thatHSDT has an error around 10% for borh deflection and stresses at
a/h :- 4. The prediction can be improved as the aspect ratio of the composite laminate in-
creases. In additions, the results obtained by GLPT are less accurate than ISSCT while

ISCT gives excellent agreement with elasticity solutions.

3.3.4 Summary ,

Based on the numerical results presented in the previous sections, the following
summary can be drawn.

1. The importance of interlaminar shear stress continuity condition in composite
laminate analysis can be recognized from the comparison between ISSCT and GLPT‘,
shown in Tables 3.11 and 3.12. In bath theories, a cubic displacement field within each
layer is used though only ISSCT satisfies the interlaminar shear stress continuity condi-
tions at the composite interfaces. It can be seen from Tables 3.11 and 3.12, bath theories
predict almost the same results for displacement and stresses. In Other words, they have
about the same accuracy for composite analysis. However, as the computational efi'ort is
concerned, ISSCT has degree-of-freedom 30% lower than that of GIN", shown in Table
3.10.

2. As mentioned in Chapter 2, the major difl‘erence between ISCT and ISSCT lies
in the assumption of transverse displacement w. The former varies in the thickness direc-
tion while the latter is constant through the thickness. If phrased difi'erently, as can be rec-
ognized from Equations (2.14) and (2.29), the significance is the consideration of at in the

Table 3.13 Closed-form solutions of a square (a=b) [0190/90/01 laminate

 

 

 

 

 

under bidirectional bending by using difi'erent laminate theories.
(2 .. .. - - - -
71' M “' “1 “2 ‘4 ‘5 ‘6
HSDT 1.8813 206641 +0.6253 02398 02056 $00435
- (02977)" (02299)
cm 1.9433 20.7323 206632 03297 02182 $010470
4 (0.2893) (02174)
rsscr 1.9555 10.7048 £06703 02876 02187 $00465
rscr 1.9377 +0.7216 + 0.6642 02876 02189 -0.0467
-0.6856 -06671 +0. 0459
Elasticity 1.937 +0720 +0663 0.292 0219 -0.0465
-0.684 -0.666 +0 0458
nsnr 0.7079 205433 :03863 0.1546 02629 4200264
(0.1930) (03059)
cm 0.7319 20.5606 10.3996 02225 0.3020 $00274
(0.1960) (03005)
10 18801" 0.7324 $05583 203999 0.1957 03006 $00274
rscr 0.7371 +05587 +0.4010 0.1955 03013 -0.0275
-o5591 -0.4027 + 0.0276
13111366113, 0.737 +0559 +0401 0.196 0301 -0.0275
-0559 -0.403 +0.0276
asnr 05004 10.5369 203027 0.1251 02814 200225
(0.1550) (03289)
cm 05077 205415 103071 0.1771 03288 #00228
(01564) (03272)
2° ISSCT 05078 +05411 203071 0.1563 03272 :00228
ISCT 0.5130 +05428 +03084 0.1555 03281 —0. 0230
—05432 -03088 +0 0231
Elasticity 0513 +0543 +0308 0.156 0328 —00230
-0.543 -0.309 +00230
nsnr 0.4293 10.5365 20.2697 0.1134 02887 400211
(0.1400) (0.3380)
cm 0.4297 20.5368 +02700 0.1580 03395 :00211
(0.1401) (03380)
10° 1sscr 0.4296 10.5366 20.2699 0.1400 03377 $00211
rscr 0.4346 +05389 +02710 0.1389 03388 -0.0214
-05389 —02710 +00214
may 0.435 +0539 +0271 0.139 0.339 4.0214
-0539 -0271 +0.0214

 

 

 

 

' quantity in parenthesis denotes the result obtained by equilibrium equations

 

Table 3.14 Closed-form solutions of a rectangular (3a=b) [0/90/0] laminate
under bidirectional bending by using difi'erent laminateth theories.

61

 

 

 

 

 

 

 

 

a .. - .. .-
71' my “’ “1 62 °'4 °5 60
user 26366 11.0372 10.1026 0.0356. 02722 $0.0262
(0.0309) (0.3822)
am 27800 11.1866 10.1093 0.0331 03405 +0.0278
4 (00313) (03408)
15507 28406 11.1254 10.1115 00319 03494 $00277
1501' 28215 +1.1496 +0.1085 0.0332 03494 —0.0269
-1.1041 -0.1191 +0.0281
Elasticity 2.820 +1.14 +0.109 0.0334 0.351 40269
-1.10 -0.119 +0.0281
HSD‘I‘ 0.8594 10.6923 10.0404 0.0177 02858 400115
(0.0150) (0.4298)
GLPT 0.9159 10.7320 10.0423 0.0161 0.4192 40.0121
(0.0154) (0.4190) ,
10 ISSCT 0.9178 107265 10.0424 0.0154 0.4196 40.0121
rscr 0.9189 +0.7261 +0.0417 0.0152 04198 —00120
-07256 -0.0435 +00123
1511966113, 0.919 +0726 +0.0418 00152 0.420 -0.0120
. -0.725 -0.0435 +00123
HSDT 05911 106404 10.0295 0.0147 02879 4200091
(0.0123) (04370)
cm 0.6070 106511 10.0301 0.0129 0.4345 40.0092
(0.0124) (0.4341)
2° ISSCT 0.6073 10.6501 10.0301 0.0124 0.4342 $00092
1scr 0.6095 +06500 +0.0294 00119 0.4343 -0 .0092
-06502 -0.0299 +00093
Elasticity 0610 +0650 +0.0294 00119 0.434 -0.0093
-0.650 -0.0299 +00093
l-lSDT 05045 106238 10.0260 0.0137 02886 $0.006
(0.0114) (0.4394)
cm 05051 10.6241 10.0260 00118 0.4395 $0.0083
(0.0114) (0.4392)
1°° Isscr 05053 10.6243 10.0260 00114 0.4392 $00083
rscr 05077 +06244 +0.0253 0.0108 0.4393 -0 0083
-0.6244 -0.0253 NM
15111116 0508 +0539 +0.0253 0.0108 0.439 —0.0083
g -0539 -0.0253 +0.0083

 

'quandtyinparendresisdenaesmeremhobtainedbyequflibdmneqmtions

 

62
thickness direction. In order to illustrate the significance, Figures 3.14 to 3.17 present the

. N closed-form solutions obtained fi'om difl'erent laminate theories. The investigations are for

a simply-supported [0/90] laminate under cylindrical bending and has aspect ratios rang-
ing from three to 200. In these figmes, the midspan deflection, maximum inplane stress at
midspan, interfacial shear stress at laminate edge, and the transverse normal stress at the
interface of midspan are normalized with respect to associated exact solutions [6]. It
should be mentioned that the results with asterisk denote the stresses recovered from equi-
librium equations, otherwise they are calculated directly from constitutive equations.

As can be seen from these figures, all theories predict excellent results except for
the transverse normal sness ‘1 from ISCT when the aspect ratio of the composite laminate
is greater than 10. However, the results from ISCT can converge to the exact solution as
the number of layers used in the analysis increases (see Table 3.11). In spite‘of this offset
difference in “1' the results of ISCT have excellent agreement with the exact solutions
even for S<10. However, it should be noted that ISSCT can also predict very good result
for S=5. Based on the numerical analysis, it can be concluded that ISCT rs necessary only

i.-.-

laminate is very thick, especially when <1x is of major concern, the variation of transverse
displacement with respect to the thickness needs to be considered.

3. Beside the comparison of accuracy described above, anather important aspect
needs to be considered is the feasibility of finite element analysis. Unlike ISSCT, the finite
element analysis using ISCT sufi‘ers from the element aspect ratio problem as pointed out
in, Tables 3.3 and 3.4. As the aspect ratio of the element is away from one, the results of
ISCT diverge. Furthermore, the ISCT demands 50% higher degree-of-freedom than
ISSCT. Therefore, it is believed that ISSCT is superior to ISCT for finite element analysis.
In the following chapters, only ISSCT is used to demonstrate the feasibility of using the

stress continuity theory for composite analysis.

63

 

 

 

 

 

102 ii I r 1 T V l i T T t r 'Vl ' I
1.1-1 .
J
/ISSCT. GLPT'
; 1.01 ——+ ~~~~~fW- __
> / ......... $22..
ISCT 4”

069' ’,\ ‘ d

08 [0/901

’ :1 V 1 ‘é'é'tto 2'0 1 {ofis'o'a'oiéo

5 (M!)

 

200

Figure 3.14 Normalized midspan deflections 14112.0) of a simply-supported [0190]
laminate under cylindrical bending by using difl'erent laminate theories.

 

 

 

 

104’ ' I rrfr'r r ‘r—r 'T'I‘I
163‘ d
1.2“ 0, .-
11 "+.\
b 1 a ”SOT 4
\x 8.
b GLPT' '2
1.1-1 iv, ..
‘
\ "1
4 \\ \~
\ ‘\~
“~ ~ ~-
1.0. ~ -._ _
5C7 [$337. GLPT'
.
o 9 [9/901
11 3 E a 1'0 2'0 E 6'0 80160

3 (1/11)

 

200

Figure 3.15 Normalized inplane normal stresses 6,012. 1112) of a simply-supported
[0/90] laminate under cylindrical bending by using difi‘erent laminate theories.

65

 

 

 

1.1 f 1 1— ' *7 r I I ' r I t ' '— I r r
4 d
ISCT
1.0q ) my
“a. ’po-
.41"
‘0’ ’..o‘
‘ m’ o.?0’ ’ ’ .0 ‘
\.'l I.
O ”
O I ’

 

 

 

\HSDT‘
06 [0/901
'2 4'1 8 8'10 20 '4'0'6'08'65' 200

Figure 3.16 Normalized transverse shear stresses ou( 0. 0) of a simply-supported
[0/90] laminate under cylindrical bending by using difi'erent laminate theories.
( " indicates the results are from equilibrium equations)

”z/azm

1.4

 

 

 

 

 

1.5+ -
1.24 -
1 1 HSDT‘
- 1 ~./ 1507 -.
‘ GLPT 1*
GLPT” ~31.
.\‘.. ‘..~. ......
1 .0 "' ~ iiiifit’.‘£i‘m‘W1
515501"
ISCT
0 9 [0/901
' 2 ' f ('5 '5'1'0 20 - 4'0 ' 8080100 200
3 (Mi)

Figure 3.17 Normalized transverse normal stresses 0, (U2. 0) of a simply-supported
[0/90] laminate under cylindrical bending by using difi‘erent laminate theories.
(* indicates the results are from equilibrium equauons)

CHAPTER 4
TECHNIQUES FOR LAYER REDUCTION

4.1 Introduction

As discussed in the previous chapter, difi‘erent laminate theories have difl‘erent as-
pects of advantage and disadvantage. For instance, HSDT is simple and has low degree-
of-freedom. However, its results for small aspect ratio ( S<10 ) are poor. In addition, the
calculation of transverse stresses in this technique needs to resort to the equilibrium equa-
tions. On the other hand, ISSCT and ISCT are suitable for bath thick and thin composite
laminates, and the calculation of transverse stresses can be obtained directly from consri-
tutive equations without an efl'ort. However, the number of degree-of-freedom in these
theories increases with the number of composite layer. A large number of degree-of-free-
dom can result in costly computation if not impossible. Fortunately, in most design cases,
instead of the whole stress distributions through the thickness, fi'equently only the stress
states at some particular interfaces are of interest. This indicates a possibility of combining
difl'erent theories together to reduce the computational efi‘ort while still retain the accuracy

in predicting stresses and deformations.

4.2 Fundamental Techniques

The goal of layer reduCtion is to combine the simplicity of sin gle-layer approach
and the accuracy and easiness for stress calculation of the interlaminar shear stress conti-
nuity theory. Figure 4.1 illustrates the idea of layer reduction. The original n-layer lami-
nate is reduced to a four-layer laminate. The decision of the layer reduction is dependent
on the interface where the stress State is of interest. As pointed out in the previous chapter,

every laminate theory can prediCt inplane snesses more accurately than transverse Stress-

67

68
es. Hence, the interface of interest should be retained in the layup after layer reduCtion.

Consequently, the two layers adjacent to me interested interface remain unchanged while
the layers above and below these two layers are lumped into two single layers. It can be
seen that this technique reduces the composite laminate from an n-layer one to a four-layer
one. As shown in Figure 4.1, the second and third layers remain unchanged. The material
propertiesusedinthesetwolayersareexactlythesameasthoseusedintheoriginalcase.
The determination of the material properties in the reduced layers, i.e., the first and fotn'th
layers, are proposed in the following sections. In this study, only ISSCT is used to demon-
strate the feasibility of the layer reduction technique. If ISCT is of interest, similar proce-
dure can be followed. .
1. Lumping the Reduced Layers by CLT

The first approach of lumping the material prOperties for the reduced layers is to
find an equivalent inplane Stiffnesses by CLT and an equivalent shear moduli by averaging
the shear modulus through the thickness. Due to this homogenization of the first and
fourth layers, bath the inplane stresses and transverse shear stresses are continuous
through the thickness of each reduced layer.
2. Lumping the Reduced Layers by HSDT

As can be shown in Appendix B, ISSCT can be reduced to HSDT for single-layer
laminates. Hence, HSDT can be viewed as a single-layer version of ISSCT. It then is pos-
sible to model ISSCT and HSDT with a consistent displacement field. That is, the reduced
layers can be modeled by HSDT while the unchanged layers by ISSCT. If the reduced lay-
er is modeled by HSDT, the transverse shear stresses calculated from constitutive equa- .
tions cannot be continuous at the interfaces of the reduced layer. However, because ISSCT
is virtually used in assembling the reduced and the unchanged layers, the continuity. of

n'ansverse shear stresses at their interfaces is guaranteed.

69

 

Original laminate Reduced laminate

Figure 4.1 Cross-sections of original and reduced layups.

70
3. Lumping Inplane Stifinesses by HSDT and Transverse Shear Moduli by Parallel Aver-
asins

The nansverse shear stresses calculated from constitutive equations in HSDT are
n0t continuous through the interfaces inside a reduced layer in the second approach while
the inplane stresses calculated from the first approach are continuous. These results do nor
fit the real situation since the inplane stress should be discontinuous through the thickness
while the transverse shear. stress continuous. However, by lumping the inplane stifiness
with HSDT and averaging the nansverse shear moduli through the thickness of a reduced
layer, the distributions of the stresses through the thickness of the composite laminate be-
come consistent with the exact solutions. In other words, discontinuous inplane stresses
and continuous transverse shear stresses through the thickness can be obtained from this

approach. The averaging technique for the transverse shear moduli is called parallel aver-

aging and implies
Q1, ‘ (Z (22’ 4.11/(24;) (4.13)
2,, - 1212;31:01/1249 (4.11:)

4. Lumping Inplane Stiffnesses by HSDT and Transverse Shear Moduli by Serial Averag-
ing
Same argument as proposed in the above approach except that serial averaging is

employed for transverse shear moduli, i.e.,

Q... - (Eliza/(gh/QE') (4.2a)
9,. = (th)/(th/Q,‘?) (4.2b)

4.3 Numerical Examples

The feasibility of the aforementioned techniques for layer reduction is evaluated

71
with the investigation of several numerical examples. Since the techniques only involve

the alternation of layup in the thickness direction without any change in property or geom-
etry in the x-y plane, closed-form solutions are available.

43.1 Cylindrical Bending

Consider a simply-supported lO-layer [0/90/0/90/015 laminate subjected to cylin-
drical bending, and assume the midspan deflection, the maximum inplane stress at mid-
span, and the midplane transverse shear stress at the laminate boundary are of interest. As
can be noted that these are also critical stresses for the composite laminate. The ten-layer
laminate is then reduced to a four-layer one, i.e., [Rl0]s, where R denotes the reduced lay-
ers. The normalizations of the numerical results with respect to ISSCT for the four ap-
proaches at different aspeCt ratios are presented in Figures 4.2-4.4. In these figures,
superscripts o and r represent for the results obtained from the original and the reduced
laminates, respecfively. Since HSDT is the one-layer version of ISSCT, i.e., the simplest
reducfion of ISSCT, the results from HSDT are included in the following figures for com-
parison. Figure 4.2 shows clearly that the midspan deflections from all approaches con-
verge to those of ISSCT as the aspect ratio increases. Among the four approaches, the
fourth approach gives the best results for all 5. For the inplane normal Stress as shown in
Figure 4.3, all approaches except the first one agree very well with ISSCT. The normalized
result for the transverse shear stress is shown in Figure 4.4. It should be poinwd out that
. although HSDT gives fair results in shear stress, the results are obtained by stress recovery
technique. If the constitutive equations are used to calculate the shear stress, poor result

can be expected.

4.3.2 Bidirectional Bending
The same 10+layer [0/90/0/90/0]s laminate is used again in the analysis of bidirec-

tional bending. The displacements and Stresses of the simply-supported, square laminate

72

1.1

 

 

w'/w°

 

 

 

 

[0/90/0/90/0],
0.8 r 1" FT 1 t f T r r r r r r
6 8 10 20 4O 60 80 100 200

S (l/h)

Figure 4.2 Normalized midspan deflections of a [0/90/0/901015 laminate under cylindri-
cal bending at different aspect ratios from difi’erent layer reduction approaches.

.73

 

 

 

 

 

1.2
"" HSDT
-. Approach 1
.. ”Woman 2
1 1.. Approach 3
. -.. Approach 4,
d HSDT
" 2
1.0%"
.1'.\;;,°.:::.-.-.-.11
o 0""\3
bx
5 0.9..
X
b
0.8.
0.7.1 1
~ ----------- ./_ ----------------
0.6 I .rrrrrr‘
4 6 8 10 20 - 40 60 80100 200
5 (l/h)

Figure 4.3 Normalized inplane normal stress a" (1/2 111/2) of a [0/90/0/90/0]s
laminate under cylindrical bending at difi'erent aspeCt ratios from difl‘erent layer

reducfion aPPmaChcs,

74

 

1.2

1.1d

”Hr/"112°

 

 

 

[0/90/0/90/0],

 

0.8

6 8 10 20 4'0 - 6030100 200

S (l/h)

Figtn'e 4.4 Normalized transverse shear Stress 63(0. 0) of a [0/90/0/90/0]s laminate
under cylindrical bending at difi'erent aspect ratios from difi'erent layer reduction

approaches.

75

subjected to bidirectionally sinusoidal loading are investigated. Since HSDT needs to re-
sort to equilibrium equations to calculate the transverse shear Stresses, it is difi'crent from
Other approaches and is dropped out from the discussion. Furthermore, since the biggesr
difi'erence among the difi'erent approaches appears at small aspect ratio, only the results at
S=4 are presented herein.

The normalized inplane displacements through the thickness are shown in Figures
4.5 and 4.6. The solid lines represent for the results obtained by the original 10~layer 1am-
inate with the use of ISSCI'. It is clear that the displacements predicted by the layer reduc-
tion techniques are continuous though the thickness. Figures 4.7, 4.8, and 4.9 present the
normalized inplane Stresses, 6“, a, and 6'”, respectively. As pointed out in the cylindrical
bending case, the first approach predicts continuous inplane Stress distributions through
the reduced layer which obviate from the exact disnibution significantly. The normalized
results of the transverse shear stresses are shown in Figures 4.10 and 4.11. The second ap-
proach gives a discontinuous shear stress through the reduced layer due to the use of
HSDT and conStitutive equations. In general, b0th the third and fourth approaches show
good approximations through the thickness.

4.4 Discussions

Of the four approaches and the numerical examples presented in the previous sec-
tions, the third and the fourth approaches seem to give better results for both displacement
and stresses. Bath approaches also give the same trends of inplane and transverse stresses
- as the elasricity solutions.

As can be seen in Figures 4.2 to 4.4, HSDT is accurate for laminates with aspecr
ratio greater than 10. However, as mentioned in Reference [33], if the stress State near the
free edge of the laminate is of concern, the stress recovery technique using equilibrium
equations is not satisfactory. On the 0mm, as will be seen in the next chapter, the inter-

laminar Stress continuity theory gives very good descriptions of displacement and mess

76

 

 

— Original
.... Approach 1
, 0'4" ~-- Approach 2
‘ ‘4 2 Approach 3
ori inal \‘.~
9 “K, 0.3J " ”pm” ‘
1 \
:5 8.23123
|-'
4 5'.
41.9.1.
'-.'4'-f2' '012'014'
"‘0
"’°‘ ‘ "1'3“ 0(011/2)
1 I
-3. g .l
z/h ‘ i
-5... ‘x‘
"3‘3? .
a-b, o/h-4 ’1“ "
[0/90/0/90/0]a

 

 

 

 

Figure 4.5 Normalized inplane displacement 17(11/2. 0) of a square [0/90/0/90/0]s
laminate subjected to bidirectional bending at S-4 from difi'erent layer reduction
approaches.

 

 

  
 

 

' 0T2 ' 0:4 ' 0:6 ' 0:8 '

Wa/Zfl)

 

 

 

 

-1.
z/h
-.3-
a-b, a/h-4 "4"
[0 /90/0/90/0],

 

Figure 4.6 Normalized inplane displacement 6(0, b/2) of a square [0/90/0/90/0]s

laminate subjeCted to bidirectional bending at S=4 from difl‘erent layer reduction
approaches.

78

 

---- Approaehl
--- Approach 2
Approach 3

 

 

0.1 -

 
  

 

' 48 ' -f6 ' -.4 -f2 ' 032 ' 014 - off 018 '

’X(°/2¢b/2)

 

a-b. a/h-4
[0/90/0/90/0],

 

 

 

 

Figure 4.7 Normalized inplane normal stress 51(4’2' b/2) of a square [0/90/0/90/0]s

laminate subjected to bidirectional bending at S=4 from different layer reduction
approaches.

79

 

 

 

 

 

 

 

 

— Original

on. Approach ‘

--- Approach 2

0.. Approach 3

-- Approach 4

fi-fa f—is ' -'.4 - 42' ' 012 ' 014 Cafe ' 0:8 '

ay(o/2.m)

a-b. a/h-4
[0/90/0/90/0].

 

 

 

 

Figure 4.8 Normalized inplane normal Stress 6, (tr/i b/2) of a square [0/90/0/90/0]s
laminate subjeCted to bidirecdonal bending at S=4 from difl'erent layer reduction
approaches.

80

 

— Original
Approach 1
Approach 2
Approach 3
Approach 4

 

 

   

a-b, a/h-4
[0/90/0/90/0]s

 

 

 

 

 

Figure 4.9 Normalized inplane shear Stress 51, (0, 0) of a square [0190/0/90/01s laminate
subjected to bidirectional bending at S=4 from difi‘erent layer reduction approaches.

81

0.5 -

 

 

 

 

0:4
5x2(0,b/ 2)

a-b. a/h-4
[O/SO/O/SO/OL

 

 

 

Figure 4.10 Normalized transverse shear stress in (0, b/2) of a square [0/90/0/90/0]S

laminate subjected to bidirectional bending at S=4 from difl'erent layer reduction
approaches.

82

 

 

 

 

 

 

 

[0/90/0/90/0],

 

Figure 4.11 Normalized transverse shear Stress in (tr/2. 0) of a square [0/90/0/90/0]s

laminate subjected to bidirectional bending at S=4 from difl'erent layer reduction
approaches.

83
near the free edge. Therefore, it is believed that ISSCT is superior to HSDT in this respecr.

AnOther technique in reducing the computational efi'ort can be achieved by com-
bining ISSCT and HSDT together. This technique implies the use of different types of ele-
ments in strucntral discretization. It is suggested that the ISSCT will be employed only
where theStressStateisofintereseandtheremainingpartoflaminatecanbemodeledby
HSDT. This mixed mchnique can reduce the total number of degree-of-freedom consider-
ably, especially for the composite laminate with many layers. However, special attention
should be paid to the compatibility between the two difl‘erent theories. Since this tech-
nique is very similar to the n'aditional finite element analysis for a complicated structure,
e.g., a structure composed of truss, beam, and plate substructtues, it is not included in this

Study.

CHAPTER 5

APPLICATIONS OF ISSCT IN
VIBRATION, BUCKLING, AND NONLINEAR ANALYSIS

5.1 Introduction

In Chapters 2 and 3, the derivations and assessments of ISCT and ISSCT are
accomplished. Static bending is used to demonstrate the merit of these new theories. It is
concluded that for very thick composite laminate (8(5) and for very high accuracy ISCT
is necessary. Otherwise, ISSCT is more efficient for computation. In order to further
inveStigate the applicability of ISSCT for engineering analysis, the governing equations of
laminated Structures in natural vibration, buckling, nonlinear bending, and nonlinear
vibration are obtained Some numerical examples need to be solved and compared with
elaSticity solutions to assess the new theory. Moreover, the feasibility of using ISSCT for

free-edge analysis is presented.

5.2 Natural Vibration
' For a composite laminate with some particular boundary conditions, linear free
vibration analysis gives the resonant frequencies and associated mode shapes. These kinds
of information provide a valuable insight into the Structure performance under dynamic
loading. Hence, in this section, the governing equation for linear, undamped, fiee vibration
will be derived. In addition, some examples will be examined to justify the accuracy of the
laminate theory. .
The Lagrangian of a teetangular composite laminate with dimensions of a x I:

under free vibration can be written as

84

85

r
21 o ‘ a T 2:
. C b 2 - ”
L 10,101.52 '5 , IL, } {28"}‘1‘4’4‘
2 6 x2 x:
x) x)
5 l
-I3I3I2h§p(liz+92+wz)dzdy¢t (5.1)
‘2

where thefirsttermisthestrainenergystoredin the structurewhilethe secondtermisthe
kinetic energy associated with the time-varying response. In addition, C) denotes time
derivative and p stands for mass density. By using constitutive equations, the Stresses can
be substituted by strains. Employing the strain-displacement relations and carrying out the
integration through the thickness, the Lagrangian then becomes

.: T :
L . g j; [3118.171310] 18.} + {8.171001 {2.15— {11,} [mar {101-whoa (52)

In the above equation, all the norations used in Chapter 2 are followed. Some new nota-
tions are defined below,

1517.1 - 2 (1:49") (~§°)’(~§°)4z) (5.3)
fill
[”50] ' .1 0 82 0 83 0 42284 A2184 32284 521.4 (5.4)
0 1’1 ° 1’2 ° 83 A1294 51184 31294 51194

m a 2 pm (21" 2‘; ') (55)
1'- 1
To satisfy the shear traction free conditions on the laminate surfaces, the same can-
Straint matrices used in Chapter 2, i.e., Equations (2.37a,b), can be inn'oduced into Equa-
tion (5.2). The Lagrangian then results in

86
. . - - - . .- T - .' _
1 - g j; [3118.1’1340 (11,} + 18.1’1580 {11,} - (11,) 154421 (11,) «13214144 (5.6)

In drecaseofclosed-form solution.Since there exists anexactmodal function for
thispudcularpmblemthedisplacementmauicescanbeassumedas

ti.) - 10.112)!“ (5.7a)
{is} - 10,1127)!“ (5.71))

where [0.] and [DJ are matrices containing assumed modal functions and their deriva-
tives with respect to the inplane axes. The matrix {5?} consists of unknown magnitudes of
the displacement variables of the particular mode shape. In addition, 1"“ represents for the
timevariantpartofthe displacementrunctions whereja- J3 and o isthe circularfre—
quency. By plugging Equations (5.7a,b) into Equation (5.6) and using the Lagrange’s

equations
%%)-%; a {0} , i a I, 2, ....,4n+1 (5.8)

the governing equation for this eigenvalue problem can be obtained, i.e.,

101 (ii-02101 11?} - {0} (5.9)

In Equation (5.8), f,- is the i-component of the column mauix {1?} .

As the static bendingsexarrlinedinChapterLfinite element analysis isrequiredto
study the Structures with general geometryandboundarycondition. Forthis typeofeigen-
value problem, a set of interpolation functions are introduced

{8.} - 111.1 m 4'" ' (5.104)
{X1} -- 1111,1110!“ (5.10b)

Substituting these functions into Equation (5.6) and employing Lagrange’s equations,

Equation (5.8), the following finite element equation for a Single element can be achieved.

87
110 m «13140120 =- 10} (5.11)

where [K] is the same stifiness matrix as used in static case while [M] the consiStent mass
matrix associated with the assumed interpolation functions.

Once the governing equation for the modal analysis is obtained, several examples
are used to demonstrate the acctnacy of ISSCT in vibration analysis. The fundamental fre-
quencies of Simply-supported [0], [0190], and [0/90/0] laminates with aspect ratio ranging
from fan to 200 under cylindrical bending are [shown in Figln'e 5.1. The same material
properties as used in Chapter 2 are examined in the simulation. In this figure, the exact fre-
quencies [28] are obtained using two-dimensional elasticity analysis while the ISSCT
results are obtained from finite element analysis using fotn' layers and four elements. It is
clear that the ISSCT results agree very well with the exact solutions in both thin and thick
composite laminates. '

Table 5.1 presents the normalized fundamental frequency of a simply-Supported
- [0190/90/0] square laminate with aspect ratio a/h=5. Difi‘erent anisotropic ratios, 5,15, , for
the material is also investigated and compared with three-dimensional elasticity results
[32]. In the finite element analysis, because of laminate symmetry, only a quarter of the
plate is examined. It can be seen that with a 4 x4 mesh, the finite element solutions con-

verge very well to the closed-form solutions.

5.3 Critical Buckling Load

For a composite laminate subjected to inplane loading, the critical buckling load is
the essential information for stability analysis. The buckling phenomenom occurs due to
the coupling between the applied inplane loading and lateral deflecuon. In this study, the
principle of minimum potential energy is used. The total potential energy of a rectangular
laminate subjected to uniformly compressive and shear loads along its boundaries can be

written as follows [36], i.e.,

88

 

 

   

[0/90/0]

[0/90]

 

10,! '(ph/D")‘/'

 

 

rr r f

8 10 20 4b 'GTO'B'O'ii'JO

s (4/11)

200

Figure 5.1 Fundamental frequencies of simply-supported [0], [0/90] and [0/90/0]

laminates with different aspect ratios under cylindrical

Table 5.1 Normalized fundamental fiequency 1.. of a simply-supported [0190/90/01
square laminate.

 

 

 

 

 

 

 

 

Vnsvuavzs-IOZS

5 IE Elasucrtyl' ' 32]
' 2 Cloud-form FEM(4x4)
40 10.752 10.698 10.724
30 10.272 10.228 10.255
20 9.5603 95310 9.5604
10 8.2103 8.3366 8.3679
3 6.6185 6.7062 6.7354
2
on p . 3
2., T E; , a b 5 , h 1

£2 a 53 :- 1x lO‘psi;Gn a 613 :- 0.6x 1061750623 2 0.5x 106psr'

 

I a b 2 - it ’3

" 10101.? a: '11 +{ an} { a} “”4""
a
I) ‘7

h
, - 2 2 2
1.112.243 3 +3144
5 1 an 2 av 2 aw 2
{3335511531 +137) +153) 142414:
2

It
a b 2 314311 311311 319319
-I0IOI_ff”(8;8_y-+§E_y'+§;5;'d2dydx
2

h It
5 ‘ -
.. 10.1.21: 0,111,. a 4,115-0) dzdy - I”; (501, _ b ..f’vl’_ 0) 4,4,
2 2

h h
- Brit (from: - a ’fxyv" ' o) 42", - ”Pb 017"" " b-fxru" " 0) ‘1de (5.12)
.2 -2

Intheaboveequation,thefirsttermisthestrainenergy storedinthe structureThesecond.
third, and forum terms are the coupled potential energy components due to inplane loading
fpf, andfx, respectively. The final forn' terms are the potential energy of external forces
' exerted on the boundaries. Since these four terms correspond to the inhomogeneous term.
i.e., the force vector, in the final eigenvalue equation, they are not relevant to the calcula-
tion of the homogeneous eigenvalue problem. Therefore, they are omitted in the following
derivation. In addition, for simplicity, only the compressive loading f, is considered
herein. The terms associated with f, and f” are removed from the total patential energy.
Following the notations used in the previous seetion, the displacement field for a

composite layer and its derivative can be written as

91

(0

t :1 - 11191185") (5.13)
a‘ (0

3'1? , a (0 (0

at am, 112. 1 (5.14)
3}

SubStituting these expressions into the pctential energy and manipulating the Strain energy
term as in the vibration study, the total potential energy after integration through the thick-

ness becomes

1 - 21313 1 {8.1715120 18.11-18.171380 11?.) - .53., {8318503318.}

aw 2
1.413;) 1414: , (5.15)
where
135.] - 2 (r1. 1N§”1'1N§°1421 (5.16)
i-l

is the assembled matrix through the thickness.
The closed-form solution for the critical buckling load is also valid for the prob-
'lems with simply-supported boundary conditions and cross-ply layup. The buckling mode

shape in the x-y plane can be assumed either a sine or cosine function with unknown mag-

nitudes, i.e.,
{id's- 10,111?) (5.16a)
ii.) . 10,111?) . (5.16b)
and
33302:} - (gt-(DJ) {17} (5.160)

Again, {if} consi5ts of unknown magnitudes of the displacement variables. Then, by
employing the principle of minimum total p0tential energy,

811 'e 0 - (5.17)

92

the homogeneous eigenvalue equation can be obtained

101 {fl-1,1811?) =- 0 (5.18)

In a similar way, a finite element equation can be derived for a more general analy-
sis. Instead of the exact mode shapes as assumed in the closed-form solution, a set of inter-
polation funcfions in the x-y plane is introduced in the finite element method.

{12.} =- [11,] {X} - (5.19a)
ti.) - 01,118} (5.19b)

Substituting these interpolation functions into Equation (5.15) and employing the princi-
ple of minimum potential energy, the finite element equation in terms of the nodal dis-

placement variables can be obtained A
m {Xi-210.111!) = o (520)

The normalized firSt buckling load of a simply-supported [0/90/90/0] square lami-
nate with aspect ratio a/h=10 is Shown in Table 5.2. Because of the symmetry of the rect-
angular laminate, only a quarter is required in the finite element analysis. The quarter
laminate is discretized into 16 equal elements. The results from both closed-form solution
and finite element analysis are presented with difl'erent anisou'opic ratios along with three-
dimensional elasticity solutions [33]. Similar analysis is performed on an asymmetric [0/
90] laminate and the results are given in Table 5.3. These results show that the ISSCT
analysis yields satisfactory predictions for b0th symmetric and asymmetric laminates. For
anorher asymmetric laminate [0/90/0/90/0/90]. Table 5.4 presents the closed-form solu-
tions. Comparing these results with those obtained from elasticity analysis, it is clear that

ISSCT predicts the buckling loading of the first mode very accurately.

93

Table 5.2 Normalized first buckling load 14, of a simply-supported [0/90/90/0]
square laminate.

 

 

 

 

 

 

 

 

 

s n: my [331
‘ 2 Closed-form FEM(4x4)
40 22.8807 23.1262 23.1360
30 193040 195545 195385
20 15.0191 152759 152198
10 9.7621 10.0283 9.9038
3 52944 55957 53707
3
ing—:7; aabslo;h=l

£2 2 E3 =- 1x 1061750612 8 013 a 0.6x 105513623 8 05x106psi

v12 3 V13 =- v23 =- 0.25

Table 5 .3 Normalized first buckling load kg of a Simply-supported [0/90] square
laminate.

 

 

 

 

 

 

 

 

 

E A: film“ ' [33) 'SSC'
DC!

' 2 ty Closed-fan FEM(4x4)
40 ' 22.8807 23.1262 23.1360
20 15.0191 152759 152198
10 9.7621 10.0283 9.9038

3

1.1-'51:; =b-10,h=

51"

a, e E3 = 1x10605120” - 6,, . 061410517813023 =‘05x10‘psi

v12 " v13 " v23 " 0'25

95

Table 5.4 Normalized first buckling load 1., of a simply-supported [0/90/0/90/0/90]
square laminate.

 

 

 

 

 

 

 

 

. . ISSCT
E /E Elmer 33
' 2 ‘7': ' Closed-form
40 23.6689 23.6673
20 15.0014 15.0126
10 9.6501 9.6289
c3
3.32;? asb=10;h=1

£2 3 E3 = 1x 1062113612 a G1, a; 0.6x 10612.12;st = 0.5x lO‘psi

vu-vu-vn-OJS

96

5.4 Nonlinear Bending

As pointed out in Reference [36], the composite laminates containing the coupling
efl‘ect between the transverse deflection and inplane force are more sensitive to the nonlin-
ear efi'ect. Even in the range of small deformation defined by conventional analysis, the
laminate can behave in a nonlinear fashion. Many investigations have used difi'erent lami-
natetheoriesanddifl'erenttypesofnonlineafity [2436-39] forthis Snldy.Inthissection.a '
laminate subjeCted to moderately large deflections is examined with the use of interlami-

nar Shear stress continuity theory.

5.4.1 Formulation of Nonlinear Equation

The material is assumed to behave linearly and elastically though the linear rela-
tionships between Strains and displacements are no longer valid. In fact. a nonlinear rela-
tionship of vonKarman sense is considered, i.e.,

3141314r

2, 8v 1 3w 2. . 314 311 awaw
8: ‘ 3732‘s? 1 er ' art‘s) - 2‘» 131-01321;
BV 3111. 3a Div
2"» ' it"s? 2‘» " 8'33; (5'21)

Itcanbe n0tedthatthe expressionsforthetlansverse shearstrainsremainthesameaslin—
ear case while the inplane strain components are modified with quadratic terms which
involve the first derivatives of transverse displacement component. Since the reduction of
displacement variables from Equation (2.27) to Equation (2.28) is achieved by imposing
the shear Stress continuity on the interface, this manipulation remains the same for borh
linear and nonlinear analysis. Thus, following the linear analysis and Equation (5.21), the

strainsineach layercanbewrittenas

(0

ex

1, =1~§°112§"}+1NN,11X,} (5.224)
28‘, - -

{ :22} .- [N59] {119} (5.22b)

where

'13w 0
'23;
[N . ‘3‘“ .
1319 law
_25'y' 252

 

 

319

= 8';
{X,} aw . (5.24)

33

The same notations defined in Equations (2.31) and (2.32) are also used. It should be n0ted
that [NNL] is a function of the derivatives of transverse displacement. It constitutes the

nonlinear part of the analysis.
Again, the principle of virtual displacement is employed for deriving the govern-
ing equation. Substituting the Stresses by suains of Equation (2.29), plugging Equations

(5220b) into Equation (2.29), and integrating through the thickness yields
[3131183071580 {2.} + {5311'13811 {8.} 1160171881“) (8,}

+ (88.17138...) (8.1+ 1“,}7138'01121 {1,1 -qu 1414: = 0 (5.25)

In the above equation, the following notations are used to denote the assembled matrices

through the thickness,

tskrttl . 2 (Returning?) 1N£°ldzl (5,25,)
i-l

[skate] =- 2 (.12-. IN? 1'12:o l [N111] dz) (5.26b)
1'81 -

98

[Sk‘ml ' 2 (£8-12[NNL]T[Q:D] (”141.1“) (5.256)

in 1
The introduction of vanished Shear uactions on bOth top and bonom surfaces of the
laminate results in the reduced displacement vectors,

{8.} - [5,) 1i.) (5.27a)
18.} - 15,118.} (5.27b)

These are the same matrices as used in Equations (2.37a,b). Substitute the reduced mani-

ces into Equation (5.25), the principle of virtual displacement becomes,

Isl: (15311155701 {12,} + (851.100,) 15?.) + {58.171800 (55.1

+1si.1’tsft~tzl {8.1+ 188.171.1803] {101 -48... 1414: - o (5.28)
in which
(sim . 15.17158.) 15,1 (529a)
[sic] - 15.171501 15,1 (5.29b)
[sim] - 1381111115,] (5.29c)
[sim] - [christian] (5293))

As in the linear case, the following interpolation functions in an element are

assumed,

if.) - 111,10!) (5.30a)
ti.) . 111,111!) (5.30b)
{101 - 111.1123} (5.30c)

w - 110 m (5.30d)

where {X} is the nodal displacement vector while (v.1. [‘11,]. [111,] . and [w] are the
interpolation functions corresponding to the displacement vectors. By using these interpo-

99
lation funcrions, the principle of virtual displacement, Equation (5.28), leads to the follow-

ing finite element equation,
([K] + [KNL(W)]) {X} '- {F} (5.31)
where
m - [3]; ( {v.JTISkd {v,} + {wrists {)5} max (5.32)
[Irmwn - m; < {v.17tskml by.) + {“176ka w.)
+ {v.fllskml at.) Web: (5.33)
{F} .. m; tvlqdydx (5.34)
It should be noted that the linear part of the stiflness matrix and the external loading veCtor
are the same as those derived in the linear analysis in Chapter 2. The major difi'erence
between the nonlinear and linear studies is the nonlinear part of the stifl'ness matrix which

is a function of the transverse displacement.

In the solution phase of the nonlinear governing equation, a standard Newton-

Raphson method is used. First, the governing equation is rewritten as
{f} 8 ([10 + [KNLll {X} - {F} = {0} (5.35)

Then the Jacobian can be calculated. The component at ith row and jth column of the Jaco-

bian matrix is defined as

a".
J" ' 55; (5.36)
where the subscripts of the column vectors represent for the corresponding components

accordingly. Once the Jacobian matrix is formulated, the numerical iteration scheme fol-

lows. A brief analysis is given below,

m "" =- (m + [M {X} “"m {X} “"3 {F} (5.37a)

100

m {X} ""n {A X} “" -- -{n "" (5.371»

{X} W" = {X} "" + {A X} “0 (5.37c)

where the superscript It denotes the result of kth iteration. The first iteration starting from
the null nodal displacement vector gives the linear solution of the equation. As the itera-
tions continue, the analysis is assumed to converge when the successive change of the dis-
placement is less than 0.1%.

In the following sections, several examples are used to examine the feasibility of

using ISSCT for nonlinear bending.

5.4.2 Laminates Subjected to Transverse Loadings

FirSt, a pinned-pinned [0/90] laminate under uniform loading over the entire span
is studied. This problem was investigated by CLT [36] for aspect ratio equals to 225. For
such thin composite laminate, the transverse shear effect can be negleCted. Therefore, the
ISSCT is expected to yield a result close to CLT. Figures 52a and 5.2b present the mid-
span defiecrion and the inplane force resultant at difi‘erent loading magnitudes. The mate-
rial properties used are the same as those in Reference [36], i.e.,

EL a 20 x106psi.ET = 1.4 x 105px}, cu a a” = 0.7 x 10%.“; v“. = 0.30

The ISSCT results are obtained by using four layers and four elements for finite element
analysis. The dashed lines in Figures 5.2a and 5.2b represent for the linear results calcu-
lated from the same ISSCT model. It is clear that nonlinear analysis from ISSCT coincides
with with that in Refernece [36] while the linear analysis erroneously predicts bOth the
midspan deflection and the inplane force resultant. It should also be noted that the struc-
ture behaves differently for upward and downward loading. This is due to the inplane
forces caused by the couplings of asymmetric layup and geometrical nonlinenrity.

One advantage of using ISSCT is the simplicity and accuracy in the calculation of

101

 

 

 

 

[O/m] ‘ \“‘
#9“, hill-0.04"

 

 

 

 

(a)

 

 

 

 

’ - [also]
#9“. h-0.04"

 

 

 

 

(b)

Figure 5.2 Pinned-pinned [0190] laminate with aspectratio S=225 subjected to uniformly
distributed loading : (a) inplane force resultant; (b) midspan deflecnon.

102

transverse shearstresses. Thisisalsotrue forthe stressesfromnonlinearanalysis. Figures
5.3aand5.3bprecentthemaximuminplanenormalstressatthemidspanandthetrans-
verse shear stress at the midplane of the laminate edge, respectively. Figure 5.3a shows the
inplane normal stress obtained from linear analysis my over-predict or underpredict the
actual sn'essdependingupontheloadingnngeThesameobsavafionappfiesmtheuans-
verse shear stress, shown in Figure 5.3b. Moreover, it is interesting to see that the trans-
verse shear stress at the interface even changes its sign as the loading is higher than a
certain value, 751b/in in this example. These unusual results can become a very important
issue in composite design and need to be carefully examined.

The stress distributions through the thickness for the same locations as shown in
Figures 5.3a and 5.3b are given in Figures 5.4a and 5.4b. In these figures, the stress distri-
butions at different deflection levels are presented. It is seen that the profiles of the inplane
stress distribution remain the same at difierent loading levels, however, those of the trans-
verse shear stress alter dramatically as the loading increases. The nonlinear analysis gives
a tremendously different stress state than the linear analysis and can result in a completely
difl'erent prediction for failure mode. '

As mentioned in a previous paragraph, the unusual nonlinear behavior of the struc-
ttn'e arises from two inplane forces caused by the couplings due to asymmetric layup and
geometrical nonlinearity. And it is known that the magnitude of the inplane force depends
on the boundary conditions. Therefore, it is interesting to study the efl’ect of difi'erent
boundary conditions on the nonlinear structural behavior. Figures 5.5a and 5.5b give the
. normalized midspan deflection and coupled inplane force resultant as a function of trans~
verse deflection. The composite laminate is of [0/90] and is subjected to a uniform load-
ing. Three different boundary conditions are of interest. The subscripts L and NI. denOte
the results fi'om linear and nonlinear analysis, respectively. Among the three boundary
conditions studied, i.e., pinned-pinned. pinned-clamped. and clamped-clamped. the

pinned-pinned one gives the most significant nonlinear efi'ect and should recieve more

103

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[0/90]
4-9", h-O.4" ‘
(a)
— nonlinear 1% “ --
--- linear ‘ 75.
”ta/I‘ll 50-
25..
'-130'-1'00r ~50 fl 1 5'0 100 ‘ 150 j
q (lb/in) -25-
-50..
“75‘. {also}
:=— -— ----- _ - b9", h-O.4"
(b)

Figure 5.3 Normalized stresses of a pinned-pinned [0/90] laminated with $222.5 sub-
jected to uniformly disuibuted loading : (a) o, ( (12,-h/2) : (b) (3,,3 ( 0, 0).

104

 

 

 

 

 

 

 

 

 

 

 

 

 

I ll .
o 4 ' —
. d ‘ --- e/h-OJQI
I - ° I/h-O.‘75
z/h 05‘ y -— aha-0.675
0.2- ,'
0.1 -
if 3606 1400' - V Wr’ado' - -
-4 ”xx/hi
,2;
{:4 ’ --4 [0/90]
1;“ b9“, h-O.4“
(a)
' ' . e - “Kn, — Linear
’o “4" \‘g‘ '" 'M191
" t \‘s. ‘ ’ VIN-0.329
. z/h “(3‘ 1,: -- I/h-O.475
‘ ‘ ~ 0.2. ‘0' - I/h-OJQB
‘~.~W\ '0’ .'°° 'Mm
O. ."'.’ \
o’ I \ ‘ - - .
-53 .30 -30 -22», \2r ‘. 5'0."
_.r ’I ”l
0.2/IQ] N .s‘. \ \ ' ”l’
- d ‘e I ”’
'2 g. \ ’ I
‘53“ E o/
[0/90] --4* lo
41-9". III-0.4“
(b)

Figure 5.4 Normalized stresses of a pinned-pinned [Ol90] laminate with S=225 sub.

jected to uniformly distribuwd loading : (a) a, ( 112, z) ,' (b) on ( 0. z).

 

 

105

 

‘3 7‘
an N

P
.a.

annjnanlnna

 

 

wn/h

 

1 a - a clamped-damped
- a-a clamped-pinned
_1 2 ‘ e—e planed-pinned

 

 

b9“, h-0.04"

 

(a)

 

 

van/h
P
on
W
+
.

 

[0/90]

 

‘ b9", Ira-0.04"

 

 

0))

Figure 5.5 Normalized nonlinear results of [0/90] laminate with S=225 subjected to
uniformly distributed loading in three difierent boundary conditions : (a) midspan
deflections; (b) inplane force resultants.

106

attention. In addition, it is interesting to see that in the clamped-clamped boundary condi-
tion, even the stacking sequence is asymmetric, the laminate behaves like a symmetric
one. In Other words, the direction of transverse loading does not change the magninrde of
deflection. This is believed to be due to the vanished inplane force in the [0190] layup [37].
Since the clamped-pinned boundary condition has the intermediate coupling force
betweenthetwoextremecasesitbehavesasacompromiseofthose two.

Once the cylindrical bending is examined. it is to investigate bidirectional bending.
Asquare[O]laminatewithaspectmfioof100isofintm'ectlhecompositelannnatehas
the following material properties

EL/ET 8 3.0, GLz/ET 3 0.5, VLT 3 0.25.

It is clamped around four edges and is subjected to a transversely uniform load. Because
of the large aspect ratio, the laminate is analyzed by CLT in Reference [38]. The load-
deflection curve is shown in Figure 5.6. The solid line represents for the result of normal-
ized central defiecrion obtained by pernubation method in Reference [38]. The open cir-
cles are the results of ISSCT using quarter laminate and a 4x4 mesh. It is obvious that

these two predictions compare very well with each other.

5.4.3 Laminates Subjected to Inplane Loading:

All the examples shown above are the laminates with transverse loading. The same
analysis can be performed for strucnrres with inplane loading. The same [0/90] laminate
with pinned-pinned boundary condition is subjected to inplane compressive loading. The
ISSCT result is based on finite element analysis. It is shown in Figure 5.7 with that
obtained from Reference [36]. Good agreement is concluded. Besides, it should be noted
that the linear buckling load obtained from linear analysis gives the upper bound of the
load—deflection curve.

Similar analysis is performed for a simply-supported [0190] square laminate with

107

 

uq,a,‘/E, h‘

 

 

 

 

[0]
Ema, - 3.0
Gu/E, - 0.5
v, - 0.25
0.0 0.4 of: 1T2 1T5 2.0

w./h

Figure 5.6 The load-deflection me of a square [0] laminate with all edge clamped and
alh a 100 is subjected to uniformly distribumd loading.

108

 

 

5
— M130]
4 a m
4‘
linear buckling load
3_.-------------------.° ......................................

 

 

axial compressive load (lb/in)

 

 

 

[0/901
b9",h-0.04"
0 r r V r V T ' I '
0.0 0.1 0.2 0.3 0.4- 0.5

midspan deflection (inch)

Figure 5 .7 The load—deflection cave of a simply-supported [0/90] laminate under
cylindrical bending with S=225 is subjected to inplane compressive loading .

109
aspect ratio of 1000. The material properties used in the simulation are as follows.

51 8 25001’0. £2 =- EfIZOGPa, 612 =- 6”: lOGI’a. G” a 4GPa, v12 = v13 3 V23=025

Figure 5.8 presents the load-deflection curve of the analysis. A 2x2 mesh is used
for a quarter of the laminate. Unlike the laminate under cylindrical bending, the laminate
under inplane compression does not buckle as the load increases. The linear buckling load
asindicatedinthediagramisnomorethanasmalldeformation.

5.5 Large-Amplitude Vibration

f The normal mode phenomenon has been shown to occur for beam and plate Struc-
tures in large amplitude vibration [41].. It is also presented in Reference [42] that for com-
posite laminate with aspect ratio greater than five and is subjected to nonlinear vibration
with amplitude-to-thickness ratio close to one, the nonlinear analysis using vonKarman
nonlinearity can provide satisfactory results compared to those using full nonlinearity.
Therefore, by combining the nonlinear Stiffness matrix obtained in the previous section
and the consistent mass matrix established in Section 5.2, the governing equation of the

undamped eigenvalue problem for amplitude-dependent vibration can be written as
(m + {Xmll {X} - m2 {M} {X} = {0} (5.33)

To analyze the amplitude-dependent eigenvalue problem, a matrix iteration
method [43] is used. FirSt, the eigenvalue problem in Equation (5.38) is transformed into a
standard form, i.e.,

{m + [run-1m {X} a :3,- {X} (5.39)

Then, the iteration scheme takes the following steps

{m + {Km {X} ""m" [M] {X} "" = —-1(,—,,7 {X} "‘* " {5.40)

(oz) .

110

 

  

 

 

 

8
-- Linear
e—e Nonlinear

A
z
a 4-
3 . .«
_ F ........ J ..... linear buckling load
0 l '
.2
g 0 r f f v r f T ' r— r
8 2 4 6 8
g” I. maximum deflection (mm)
o ‘ ,I
0 r
o ‘ f
C I,
.9. -4~
a.
.s l

-8

 

Figure 5.8 The loadodeflection curve of a simply-supported square [0/90] laminate with
alh =- 1000 is subjected to inplane compressive loading along the x—direction.

l l 1

In the above equation, the resulting veCtor of the left-hand side is normalized to give the
desired amplitude of vibration for a particular mode and the normalization constant related
to the reciprocal of the associated eigenvalue. The iterations continue until the conver-
gence of the eigenvalue within a preset tolerance is reached. After obtaining the first mode
frequency and mode vector, the second mode frequency and mode vector can be found
similarly. However, for the second mode, a sweeping matrix needs to be introduced to
incorporate an orthogonal constraint in between the first and second mode vectors. Details
of the procedure can be found in Reference [43]. In most situations, only the fundamental
mode is of interest, therefore, the solution for the higher modes are not pursued in this
study.

In order to verify the feasibility of ISSCT for large-amplitude vibration, the funda-
mental frequency of a thin [0/90/90/0] laminate is of interest. This composite laminate has
an aspect ratio of 100 and a pinned-pinned boundary condition is subjected to cylindrical
. bending . Figure 5.9 presents the amplitude-dependent fundamental frequency of the lam-
inate. In the ordinate, A represents for the amplitude of the funamental mode while r the
radius of gyration of the cross-section. For a rectangular cross-section r a it/ (.55) . The
results obtained by using CLT from Reference [44] is shown by a solid line. Clearly, it has
a very good agreement with those fi'om ISSCT.

As concluded in the study of nonlinear bending, the boundary conditions play an
important role in the response of laminated structure. Herein, the amplitude—dependent
natural frequencies for a [0/90/90/0] laminate under three difi'erent boundary conditions
are studied. Figure 5.10a and 5.10b show the ratio of nonlinear fundamental frequency to
linear frequency at difl‘erent vibration amplitudes for a thin (8:100) and a thick (8:10)
composite laminate, respectively. It is interesfing to see that the thin laminate in a pinned-
' pinned boundary condition shows the most significant nonlinear effect. However, the least
nonlinear effect is observed in the thick laminate at the same boundary condition. The

reverse is true for the laminates in a clamped—clamped boundary condition.

112

 

 

   

 

 

5
a ISSCT
— Redd-4]
4. o
J
3l
i
<
2l
1 J
pinned-pinned
‘ [0/90/90/0]
5-1 00
0 T f r . r - r
0.5 1.0 1.5 2.0
tie/0t...
Figure 5.9 The amplitude-dependent fundamental frequency of a pinned-pinned

[0/90/90/0] laminate with S=100 under cylindrical bending.

NB

 

 

 

 

 

 

 

 

 

 

5
thaldmud-fimud
‘DdlphMM-dflflfld
‘ thetdammfl-dmmnd
3.
< l
‘<
2i
1-1
q [0/90/90/0]
5-100 .
0-.-- -+.,.---
0.5 1.0 1.5 2.0
“ta/Wot
(a)
[0/90/90/0]
5-10
' I F - 2.0
Figtn'e 5.10 Normalized amplitude-dependent fundamental frequencies of [0/90/90/0]

laminate in three difl'erent boundary conditions : (a) S=100 ; (b) S=10.

114

Due to this difi'erent behavior between thick and thin composite laminates, it is
natural to investigate the effect of aspect ratio on the dynamic behavior of the structure.
The result of fi'equency ratio for a [0/90/90/0] laminate with fixed amplitude ratio
A/r . 2.0 ispresentedinFigmeSJlaAseanbeseenfiomthisdiagramthecharactmiso
tics ofthe laminate undergoesasignificantchangeas theaspectratioofthelaminateis
smallerthan 20. Thisresultcoincideswith the findingofthe transverse sheardeformation
efi‘ect on the composite laminate presented in Reference [6]. Therefore, there is a doubt if
the transverse sheardeformationplaysanimportantrolein theresponse ofthicklami—
nates. Beside the resonant frequencies, the information of mode shape is also crucial in
structural analysis. In Figure 5.llb, a coherence factor between the linear and nonlinear
mode shapes, We} and {9m} , is intoduced in Reference [45], i.e.,

({omli'le)’

coherence a —— -
{{qipLVme (twirling)

 

The coherence factor gives a value between zero and one. If two mode shapes are exactly
the same, it gives a value of one. A zero coherence means that the two mode vectors are .
orthogonaL Figure 5.11b shows that as the aspect ratio of the laminate becomes less than
20, the coherence factor drops sharply for laminates of all kinds of boundary condition.
This implies that the mode shapes obtained from nonlinear analysis deviates from those
fromfinearanalysis'l'hisresultmayjeopardize the assumptionofusinglinearmode
shape for nonlinear structure analysis [46].

5.6 Free-Edge Stresses

The free-edge stress has long been recognized as a unique problem in laminated
composites [47-49]. The purpose of this section is to assess the feasibility of using the
interlaminar shear stress continuity theory presented for free-edge analysis. Since constant
w through the thickness is assumed in ISSCT, i.e., the effect .of a: is ignored, a [45/45]s

115

 

 

 

 

 

 

 

 

 

 

 

 

 

‘ [0/90/90/0]
0.3 .ef- ..,A/':'2.‘°-.
0 50 100 150 200
3 (Mi)
(a)
1.25
o o pinned-phned
1: a pinned-clamped
g a clamped—clamped
«H 1.00d . e I e a e e
a 8
i; a
“-
o e
‘6 0.75«
con!
0
.E’.
o
3 050 .
g .
‘§ ° {0/90/90/01
r-2.0
ozs'f'fifir'YIr vvvrA/wfir
0 50 100 150 200
S (“*0
(b)
Figure 5.ll The change ofnonlinear fundamental freq ncy and mode shape ofa

[0/90/90/0] laminate at the vibration amplitude A/ r = 2.0 ; (a) fmdmcnm frequency;

(b) coherence factor.

1 16

laminate which does not generate transverse normal stress due to inplane loading is taken
as an example. Figure 5.12 shows the mesh for finite element analysis. For convenience, it
is possible to examine only one half of the specimen. In addition, specifying the uniform
strain in the x-direction. as usually employed in the free-edge studies [49], is not a feasible
technique in this analysis. Hence, a uniform tensile loading is applied at the laminate ends.
However, the strain across the width is verified to be very close to uniform distribution.

Figures 5.1311 and 5.13b present the normalized displace-ent u (0, y, U2) and
transverse shear stress a“ (0. y, It/4) , respectively. It is clear that the finite element analy-
sis using ISSCl' predicts excellent results as obtained in Reference [49]. With these results
and the previous studies, it is believed that ISSCT can be used for general analysis for

laminated composites.

117

no x
{.
4' '5
0' " §
b-2h 0 ‘- -) 0’0
4' 'F
‘- '5
§ “D

 

 

|< 4

Figure 5.12 Mesh layout for [45l-45]s laminate in calculation of freeoedge stress.

118

1 .00 r . . , .
o ISSCT
—- Ref.[49]

0.75 4 _

 

0.50 d

4u(y,h/2)/exh

 

 

 

 

 

(a)

2.4 r r ' I ' ' ' '
o ISSCT
—- Ref.[49]

2.0 a -

 

1.64

 

0.8 -l

au(y,h / 4) / 1 O'cx

0.4m

 

0.0 e , t .
0.0 0.2

 

 

 

 

Figure 5.13 Normalized results of a [45/-45]s laminate subjected to uniform inplane
loading: (a) through-the-width in-plane displacement u (0, y, h/Z); (b) through-the-
width interlaminar shear stress a (0. y. h/4)

CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS

6.1 Conclusions

In this study, two laminate theories based on multiple-layer approach - ISCT and
ISSCT - for the analysis of bath thick and thin composite laminates are presenmd. The
easiness of the direct stress calculation fiom constitutive equations and accuracy of the re—
sults from using Stress continuity conditions are demonstrated by some numerical exam-
ples. Moreover, the expedience in computation for composite laminates with large number
of layers is achieved by the layer reduction technique. A comprehensive investigation of
using ISSCT in the vibration, buckling, nonlinear, and free-edge analysis of composite
laminates also show a good patential of using the stress continuity theories for composite
analysis. In summary, the following conclusions are drawn:

1. Two interlaminar stress continuity theories for laminated composites are developed.
One considers the variation of transverse displacement through the composite thick-
ness and the ather assumes constant transverse displacement. The former is named the
interlaminar stress continuity theory. (ISCT) while the latter the interlaminar shear
stress continuity theory (ISSCT). These theories enable a direct and accurate calcula-
tion of transverse stresses from consitutive equations for bath thick and thin composite
laminates.

2. A simple technique is developed for finding the closed-form solutions of some particu-
lar problems such as cylindrical bending and bidirecrional bending. Since no approxi-
mation is included in this technique, the error from numerical analysis can be avoided.

3. With little modification, the multiple-layer laminate theory ISSCT can be reduced to
single-layer theory , i.e., HSDT. This concludes that HSDT can be deemed as the sin-

119

120
gle-layer version of ISSCT.

4. From the numerical examples examined in this study, it is concluded that high-order
shear deformation theory (HSDT) can be used for laminates with aspect ratios (5)
greater than 10 while ISSCT S>5. Due to its threerdimensional approach in nature,
ISCT has no limitation in aspect ratio.

5. A layer reduction technique for reducing the degree-of-freedom is developed by com-
bining the interlaminar shear stress continuity theory (ISSCT) and high-order shear
deformation theory (HSDT). This technique can give transverse Stresses at desired in-
terfaces without introducing too many degree-of-freedom.

6. The finite elements derived from ISCT have thickness the same as the composite lami—
nates. Based on the numerical examples studied in this thesis, it is observed that as the
aspect ratio of a finite element is close to one best result can be obtained. The finite el-
ement analysis using ISCT seems to sufl’er from the aspect ratio problem.

7. The applications of ISSCT for vibration, buckling, nonlinear bending, nonlinear vibra-
tion, and free-edge analyses of laminated composites show excellent results. All the
investigations indicate that ISSCT is a very promising technique for composite analy-

sis.

6.2 Recommendations
Based on the work performed in this thesis, the following studies are recommend-
ed for further investigation:

I 1. This thesis gives two accurate laminate theories for predicting both displace-
ment and stress of laminated composites. The failure analysis can be performed with the
help of these types of information. For example, the first-ply-failure or last-ply-failure
analysis can be combined with the stress continuity theories while the delamination at the
interface can be modeled with a soft and thin embedded layer or by a slip layer[50].

2. The feasibility of using the Stress continuity theories in analyzing bath global

121
andlocnmspometotcomponmhminmmasmenuonedmcmptms,hasmcommended
a patential application of these thoeries in assessing the performance of smart materials
and intelligent system which are made of composite laminates and embedded sensors and
actuators. The constitutive relations of pieao-electric crystal, shape memory alloy, electro-
rheological fluid, and optical fiber can be incarporamd into these stress continuity theories.
Inthisway, the globalresponseofthesmartmaterialandintelligentsystemcanbesimu-
latedandthesn'essstatearoundtheembeddedsensorscanbeexamined.

3. The structures with viseoelastic damping materials in both constrained layer and
extensional layer configurations can initially be analyzed with these stress continuity theo-
ries. Then, by using the specific damping capacity presented in Reference [51], the damp-

ing characteristics of the viscoelastic structures can be evaluated.

APPENDICES

APPENDIX A

[E] AND {q} MATRICES

A.l Interlaminar Stress Continuity Theory

InISC‘thhecousn'aintmatriceflEJandIEJaredefinedas

 

d

 

 

. .0000000010004 0
.0000000100440 0
..0000001004000 0
.. . . . .0000010000000 0
mam»
.OOOOIOOOOOOOOQQ
mum.»
.OOOIOOOOOOOOOQQ
mama
. . . . . .OOIOOOOOOOOOOQQ
mums
.OIOOOOOOOOOOOQQ
ee ee ee e e e e e o o 1000000000000 0
000000000000 0 01 . . . .
000000000000 0 10 . . . .
000000010004 0 00 . .
000000100440 0 00
000001004000 0 00
000010000000 0 00
000100000000nn_0300
QQ
.
mama
00100000000090.00
uuun
01000000000090.00
mums
10000000000090.00
-
a
.J
E
[

(UnelJHlJneJ)

122

123

[5,} -

 

00.
00.

P1

00.

00.

00.

000
000
001

000

000

l
-100.

0
0

0

0-10.

00.

0

0

1 1
05.’ ab’

1 1
Qi.’ aa’

0 0

1 l
als’ ai.’

l 1
0:3) 9(3)

0

-l

04

 

 

 

(Mrsel4flllae0)

124

{em-[0.0.0. ..... .00—21;)
Q33

i(13:r+l3) x I

{9,} ‘ [orator oumutorosm— 923;) yr
(1454-14) x1

A.2 Interlaminar Shear Stress Continuity Theory

In 18 SCT, the following constraint matrices are used:

100000..0000 a
010000..0000 0
000000..000-lo
000000..0000-4
001000..0000 0
000100..0000 0
000010..0000 0
[5’]. 000001..000 0 0

000000..1000 0
000000..0100 0
000000..0010 0
000000..000-lo
000000..0000-n
000000..000l 0
900000..000 01“.,mmm

 

 

(5.}-

0000000000

 

9000000000

1000000000..
0100000000..
0010000000..
0001000000.
0000000000.
0000000000..
0000000000..
0000000000..
0000100000..
0000010000..
0000001000..
0000000100..
0000000010..
0000000001..

0000000000..
0000000000..
0000000000..
0000000000..
0000000000..
0000000000..
0000000000..
0000000000..
0000000000..
0000000000.
0000000000..
.000000
.0 0000

125

000000
000000
000000

.000000

000000
000000
000000
000000
000000
000000
000000
000000
000000
000000

100000
010000
001000
000100
000010
000001
000000
000000
000000

.000000

000000

1.0cuac
luLcraoeao
ccweccnao‘

OOOOOOOOO
OOOOOOO

OOOOOO'
OGOOOOO'

OCH—cracl.
euuac_.L

l
0-

960600

OOOOOOOOO'

l
o::_

 

.-
L

(“011) I (he!)

APPENDIX B
THE EQUIVALENCE OF ISSCT AND HSDT FOR ONE-LAYER
LAMINATE

For a single-layer laminate, the displacement field of ISSCT, Equation (2.27), can
be simplified as
u a (1001+ 7002+ ”1’3 4» 11¢4

v . V001 + 30¢Z+V1¢3 +3104 (B.1)

w=w°
where
3 It2 2 1.3
$1-1 ;§(2+§) +;5(Z+§)

1 It It
0, . ';§(1+‘2')(1'§)
(3.2)

3 12 2 h3
19337.5(??? ‘piz‘l'?

1 It3 2 It?-
04'304'5) -7t(z+§)

Since there is only one layer, no interfacial shear stress continuity is enforced. However,
the zero shear uaction on top and bottom surfaces of the composite laminate should be sat-

isfied, i.e., the shear su'ains at these locations must vanish,

awO . 3W0
2822 h-TO+3-X- 80 p 283 h3tl+$ .0
2 2
Hence,
3’90
70'1'1 "3; ~ (B3)

127

Similarly
Div
303 St - 3;" (13.4)

Substituting these relationships into Equation (B.l), the displacement field becomes
3“'0

" ' ”011+”li’i‘ "2“4’5;
aw (B.5)

" " V0.1 "’ Vl’: " “2"4’330

By plugging the Hermite cubic interpolation functions (B.2) into the new displacement

 

fieldandletting
U H!
and
3w U -U
V: 23'; 2‘ It ) (B )

the new expression for displacement u in terms of new variables can be obtained

4:2 3100
u 3 110+! Vx-EIZ-(Vx‘i's; ) (3.78)

In the above expression, u. is the nridplane displacement in the x direction, while w,
relates to the rotation accounting for transverse shear deformation at the midplane. In a
similar fashion, the new expression for v can also be derived as follows,
2 3w .
4 r 0 ) (3.7!»

V I Vo+1(v,-'3-F (V,+$ )

It thenisclearthatthedisplaeementfieldforISSCI'canbereducedtoaHSDT.

LIST OF REFERENCES

LIST OF REFERENCES

1. Noor, A.K., and Burton, W.S., “ Assessment of Shear Deformation Theories for Multi-
layered Compoa'te Plates,” Applied Mechanics Review, Vol. 42, 1989, pp. 1-13.

2. Kapania, R.K., and Raciti, 8., “Recent Advances in Analysis of Laminated Beams and
Plates, Part I: Shear Efi'ects and Buckling,” AIAA Journal, Vol. 27, 1987 , pp. 923-934.

3. Kapania, R.K., and Raciti, 8., “Recent Advances in Analysis of Laminated Beams and
Plates, Part II: Vibrations and Wave Propagation,” AIAA Journal, Vol. 27, 1987. pp.

' 935-946.

4. Reddy, J.N., “A Review of Refined Theories of Laminated Composite Plates,” Shock
and Vibration Digest, Vol. 22, 1990, pp. 3-17.

5. Reddy, J.N., “A Generalization of Two-Dimensional Theories of Laminated Composite
Plates,” Conununications in Applied Numerical Methods, Vol. 3, 1987, pp. 173-180.

6. Pagano, NJ ., “Exact Solutions for Composite Laminates in Cylindrical Bending,”
Journal of Composite Materials, Vol. 3, 1969. PP. 98-411.

7. Yang, P.C., Norris, CH, and Stavsky, Y., “Elastic Wave Pro agation in Heterogeneous
Plate,” International Journal of Solids and Structures, Vol. , 1966, pp. 665—684.

8. Reissner, 13., “On Transverse Bending of Plates Including the Bfi‘ects of Transverse
ggggefomfim,” International Journal of Solids and Structures, Vol. 11, 197 5, pp.

9. Lo, KH, Christensen, RM, and Wu, E.M., “A High-Order Theory of Plate Deforma-
téiéoggsart 1: Homogeneous Plates,” Journal of Applied Mechanics, VoL 44, 1977, pp.

10. Lo, K.H., Christensen, RM, and Wu, 5114., “A High-Order Theory of Plate Deforma-
tion - Part 2: Laminated Plates,” Journal of Applied Mechanics, Vol. 44, 1977 , pp.
669-676.

11. Reddy, J.N., “A Simple Higher-Order Theory for Laminated Composite Plates,” Jour-
nal of Applied Mechanics, Vol. 51, 1984, pp. 745-752.

12. Hon g, 8., “Central Delamination in Glass/Epoxy Laminates,” Ph.D. Dissertation,
Michigan State University, East Lansing, Michigan, 1990.

13. Ambartsumyan, S.A., W translated from Russian by T.
Cheron and edited by LE. Ashton, Technomic Publishing Co., 1969.

128

129

14. Whitney, J.M., “The Efi‘ect of Transverse Shear Deformation on the Bending of Lami-
nated Plates,” Journal of Composite Materials, Vol. 3, 1969, pp. 534-547.

15. Pryor, C.W., Jr., and Barker, RM., “A Finite-Element Analysis Including Transverse
31112“; l1:27fl'ects for Applications to Laminated Plates,” AIAA Journal, Vol. 9, 1971, pp.

16. Man, S.T., Tong. P., and Plan. T.I-I.H., “Finite Element Solutions for Iaminamd Thick
Plates,” Journal of Composite Materials, Vol. 6, 1972, pp. 304-309.

.17. Spilker, aim Chou. so, and Orringer, 0., “Alternate Hybrid-Stress Elements for
Anallmysrs gfl Mglmayer Composite Plates,” Journal of Composite Materrals' , Vol. 11,
, pp. - .

18. Murakami, H., “Laminated Composite Plate Theory withjsgmved In-plane
Response,” Jounral of Applied Mechanics, Vol. 53, 1986, pp. 661 .

l9. Toledano, A., and Murakami, H., “A Composite Plate Theory for Arbitrary Laminate
Configurations,” Journal of Applied Mechanics, Vol. 54, 1987, pp. 181-189.

20. Reissner, E., “On A Certain Mixed Variational Theorem and A Proposed Application,”
Igtggnational Journal for Numerical Methods in Engineering, Vol. 20, 1984, pp. 1366-
l . '

21. Seide, B, “An Improved Amoximate Theory for the Bending of Laminated Plates,”
Mechanics Today, Vol. 5, 1 , pp. 451-466.

22. Di Sciuva, M., “Development of An Anisotropic, Multilayered, Shear-Deformable
Rectangular Plate Element,” Computers tit Structures, Vol. 21, 1985, pp. 789-796.

23. Di Sciuva, M., “Bending. Vibration and Buckling of Simply Supported Thick Multi-
layered Orthon'opic Plates: An Evaluation of A New Displacement Model,” Journal of
Sound and Vibration, Vol. 105, 1986, pp. 425-442.

24. Hinrichsen, R.L., and Palazotto, A.N., “Nonlinear Finite Element Analysis of Thick
836- at; Plates Using Cubic Spline Functions,” AIAA Journal, Vol. 24, 1986, pp.
1 1 .

25. Reddy, J.N., Barbero, EJ., and Teply, I.L.,“A Plate Bending Element Based on a Gen-
eralized Laminate Plate Theory, International Journal for Numerical Methods in
Engineering, Vol. 28, 1989, pp. 2275-2292.

26. Barbero, EJ., Reddy, J.N., and Teply, 1.1... “An Accurate Determination of Stresses in
Thick Laminates Using A Generalized Plate Theory,” International Journal for
Numerical Methods in Engineering, VoL 29, 1990, pp. 1-14.

27. Pagano, N .J ., and Hatfield, 8.1., “Elasdc Behavior of Multilayered Bidirecfional Corn-
posites,” AIAA Journal, VOL 10, 1972, pp. 931-933. .

28. Jones, A.T., “Exact Nann'al Frequencies for Cross-Ply Laminates,” Journal of Com-
posite Materials, Vol. 4, 1970, pp. 476-491.

29. Jones, A.T., “ExaCt Natural Frequencies and Modal Functions for A Thick OE—Axis
Lamina.” Journal of Composite Materials. Vol. 5, 1971, pp. 504-520.

130

30. Kulkarni, S. V., and Pagano, NJ., “Dynamic Characteristics of Composite Laminates,”
Journal of Sound and Vibration, Vol. 23,1972, pp. 127-143.

31. Srinivas, S., Joga Rao, C.V., and Rao, A.K., “An ExactAnalysis for Vibration of Sim-
ply-Supported Homogeneous and Laminated Thick Rectangular Plates,” Journal of
Sound and Vibration, Vol. 12. 1970, pp. 187-199.

32. Noor, A.K., “Free Vibrations of Multilayered Composite Plates,” AIAA Journal, VoL
11,1973. pp. 1038-1039.

33. Noor, A.K., “Stability of Multilayered Composite Plates,” Fibre Science Technology,
VoL 8, 1975, pp. 81-89.

34. Pagano, NJ., and Soni, S .R.,“Models forSnrdyingFree-Edge Efi'ects,” inm-
lish 989 edited by NJ. Pagano, Elsevier Science Pub-
er, 1 .

35 Vinson. JR. and Sicrakowski. Ri- W-
W Martinus Nijhofl' Publishers, 1986.

36. Sun, CT, and Chin, 11., “Analysis of Asymmetric Compositelaminates,” AIAA Jour-
nal, Vol. 26, 1988, pp. 714—718.

37. Whrmey, I.M., .- -
lishingCo..1987.

38. Chia. C.Y., “Large Deflection of Rectangular Orthotropic Plates,” Journal of the Engi-
nezgring Mechanics Division, Promdings of the A.S.C.E., Vol. 98, 1972, pp. 1285-
1 8.

39. Reddy, J. N., and Chao, W. C., “Large-Deflection and Large-Amplitude Free Vibrations
gf4143r217nated Composite-Material Plates,” Computers tit Structures, Vol. 13, 1981, pp.
1- .

40. Barbero, EJ., andReddy,J.N., “NonlinearAnalysisof ' LaminatesUsingA
Generalized Laminated Plate Theory,” AIAA Journal, Vol. , 1990, pp. 1987-1994.

41. Wah, T., “The Normal Modes of Vibration of Certain Nonlinear Continuous Systems,”
Journal of Applied Mechanics, Vol. 31, 1964, pp. 139-140.

42. Singh, G., and Sadasiva Rao, Y.V.K., “Nonlinear Vibrations of Thick Composite
Plates,” Journal of Sound and Vibration, Vol. 115, 1987, pp. 367-371.

43. Meirovitch, 1... MW The Macmillan Company, 1967.

44.Woinowsky-Krieger,S.,“1'he Efiect of an Axial Force on the Vibration of Hinged
Bars,” Journal of Applied Mechanics, Vol. 17, 1950, pp. 35-36.

45. Kuang, 1.1-1., and Tsuei, Y.G., ”A More General Method of Substructure Mode Synthe-
sis for Dynamic Analysis,” AIAA Journal, Vol. 23,1985, pp. 618-623.

46. Kapania, R.K., and Raciti, 8., “Nonlinear Vibrations of Unsymmenically Laminated
Beams,” AIAA Journal, Vol. 27, 1989, pp. 201-210.

 

131

47. Pagano, NJ., and Pipes, R.B., “Some Observations on the Intel-laminar Strength of
Camsuminates,” International Journal of Mechanical Science, Vol. 15,1973,
PP

48. Pipes, R.B., and Pagano. NJ., “Interlaminar Stresses in Composite Laminate- An
W Elasticrty Solution,” Journal of Applied Mechanics, Vol. 41, 1974, pp.

49. Pagano, N J., “Stress Fields in Composite laminates,” International Journal of Solids
and Structures, Vol. 14, 1978, pp. 385-400.

50. Toledano, a, and Murakami, H., “Shear-Deformable rhvo-Layer Plate Theory with
Interlayer Slip,” Journal of Engineering Mechanics, Vol. 114, 1988, pp. 604-623.

51. Ungar,E.E., andKervdeM..Jr.,“LossFactorsofViscoelasticSy stemsinTermsof
Eneggsy‘tgtsigmepts," The Journal of The Accoustical Society of Amenca, Vol. 34,1962,
PP